- Open Access
An evolutionary and functional assessment of regulatory network motifs
© Mazurie et al; licensee BioMed Central Ltd. 2005
Received: 19 October 2004
Accepted: 22 February 2005
Published: 24 March 2005
Cellular functions are regulated by complex webs of interactions that might be schematically represented as networks. Two major examples are transcriptional regulatory networks, describing the interactions among transcription factors and their targets, and protein-protein interaction networks. Some patterns, dubbed motifs, have been found to be statistically over-represented when biological networks are compared to randomized versions thereof. Their function in vitro has been analyzed both experimentally and theoretically, but their functional role in vivo, that is, within the full network, and the resulting evolutionary pressures remain largely to be examined.
We investigated an integrated network of the yeast Saccharomyces cerevisiae comprising transcriptional and protein-protein interaction data. A comparative analysis was performed with respect to Candida glabrata, Kluyveromyces lactis, Debaryomyces hansenii and Yarrowia lipolytica, which belong to the same class of hemiascomycetes as S. cerevisiae but span a broad evolutionary range. Phylogenetic profiles of genes within different forms of the motifs show that they are not subject to any particular evolutionary pressure to preserve the corresponding interaction patterns. The functional role in vivo of the motifs was examined for those instances where enough biological information is available. In each case, the regulatory processes for the biological function under consideration were found to hinge on post-transcriptional regulatory mechanisms, rather than on the transcriptional regulation by network motifs.
The overabundance of the network motifs does not have any immediate functional or evolutionary counterpart. A likely reason is that motifs within the networks are not isolated, that is, they strongly aggregate and have important edge and/or node sharing with the rest of the network.
Global interaction data are synthetically structured as networks, their nodes representing the genes of an organism and their links some, usually indirect, form of interaction among them. This type of schematization is clearly wiping out important aspects of the detailed biological dynamics, such as localization in space and/or time, protein modifications and the formation of multimeric complexes, that have been lumped together in a link. Given these limitations, an important open question is whether the backbone of the interaction network provides any useful hints as to the organization of the web of cellular interactions. A first observation in this direction is that the topology of biological interaction networks strongly differs from that of random graphs . In particular, when transcriptional regulatory networks are compared to randomized versions thereof, some special subgraphs, dubbed motifs, have been shown to be statistically over-represented [2, 3]. An example of a motif composed of three units is the feed-forward loop, its name being inherited from neural networks, where this pattern is also abundant.
Transcription factors often act in multimeric complexes and the formation of these plays a crucial role in the regulatory dynamics. In order to capture at least part of those effects, transcriptional networks may be integrated with the protein-protein interaction data that have recently become available [4–7]. An example is provided by the mixed network constructed in . The network is mixed in the sense that it includes both directed and undirected edges, pertaining to transcriptional and protein-protein interactions, respectively. The motifs for the mixed networks were investigated in .
The dynamics of motifs has been thoroughly investigated in vitro and in silico, that is, in the absence of the rest of the interaction network and of additional regulatory mechanisms [10–12]. For instance, the feed-forward loop has remarkable filtering properties, with the downstream-regulated gene activated only if the activation of the most-upstream regulator is sufficiently persistent in time. The motif essentially acts as a low-pass filter, with a time-scale comparable to the delay taken to produce the intermediate protein. Furthermore, the same structure is also found to help in rapidly deactivating genes once the upstream regulator is shut off. Overabundance of motifs and their interpretation as basic information-processing units popularized the hypothesis of an evolutionary selection of motifs [2, 13].
In electrical engineering circuits, an abundant structure is likely to correspond to a module that performs a specific functional task and acts in a manner largely independent of the rest of the network. The point is moot for biological networks. A recent remark is that some of the motifs found in transcriptional networks are also encountered in artificial random networks [14, 15], where no selection is acting. However, the lists of motifs do not entirely coincide for the two cases . A visually striking fact is that essentially none of the motifs exists in isolation and that there is quite a great deal of edge-sharing with other patterns (see  for the network of Escherichia coli). The function of the motifs might then be strongly affected by their context. The use of genetic algorithms to explore the possible structures that perform a given functional task has in fact shown a wide variety of possible solutions .
It is therefore of interest to address the issue of the functional role of the motifs in vivo, that is within the whole network, and examine the ensuing evolutionary constraints. In the following, we shall show that the instances of the network motifs are not subject to any particular evolutionary pressure to be preserved and analyze the biological information available on the pathways where some instances of motifs are found.
List and annotation of network motifs
The list includes the purely transcriptional feed-forward loop, investigated in [10–12], and its version augmented with a proteic interaction . The overall list is quite similar to that found in , with the only exception of proteic self-interactions, which were not taken into account. General information on the motifs is obtained by looking at the biological processes, molecular functions and cellular components for which the genes found in occurrences of Figure 1 motifs have been annotated (see Additional data files 1 and 2).
Let us first remark that the various instances of the motifs account for 25% of all the genes annotated as transcription factors in the MIPS/FunCat and GeneOntology (GO) databases. The annotations obtained using the former database indicate that 34% of the genes involved in motifs are annotated as involved in transcriptional regulation and 31% in direct control of transcription; and that 51% of the genes have their products localized within the nucleus.
These values should be compared to 5% of all the genes annotated for transcriptional control in either GO or FunCat and 30% of nuclear localization for all annotated genes. Another relevant remark is that transcription factors are found at 93% and 11%, respectively, of the nodes with an outgoing and an ingoing transcriptional link. That is, indeed, the expected behavior for genes in a transcriptional network. These results witness the coherence of the transcription and the protein-protein interaction datasets used for finding the motifs and the published annotations.
As for the function of the genes composing the network motifs, the list of the most represented biological processes, as annotated in the MIPS database, is as follows: 50% of the genes are involved in metabolism, 34% in transcription, 21% in cell cycle and DNA processing, 12% in interaction with the cellular environment (10% in cellular sensing and response), 10% in cellular transport and 9% in rescue/defense.
Phylogenetic profiles of network motifs
To ascertain the presence of any special evolutionary pressure acting to preserve over-represented patterns, we have performed a protein comparative analysis between Saccharomyces cerevisiae and the four hemiascomycetes Candida glabrata, Kluyveromyces lactis, Debaryomyces hansenii and Yarrowia lipolytica, recently sequenced in . The fact that the four organisms share many functional similarities with S. cerevisiae and yet span a broad range of evolutionary distances, comparable to the entire phylum of chordates, makes them ideal for protein comparisons. Details of the sequence comparisons are reported in Materials and methods.
Previous evolutionary studies on the motifs have explored the presence of common ancestors in different instances of the motifs. The upshot was that the various instances are not likely to have arisen by successive duplications of an ancestral pattern . Here, we consider a different statistic based on the phylogenetic profiles  of the genes within the motifs. The profiles are constructed considering an ensemble of organisms and looking at the co-occurrences in the compared organisms of the genes composing the interaction pattern. This is quantified by the evolutionary fragility, F i (as defined in Materials and methods), of the interaction pattern i. A small value for the fragility indicates that the genes composing the pattern tend to co-occur in the other compared organisms, hinting at an evolutionary pressure to preserve the pattern and at its functional importance. We shall compare the statistics of the evolutionary fragility for different classes of interaction patterns, thus providing a test of the evolutionary significance of the criterion of overabundance used to identify network motifs.
Function in vivoof realizations of the motifs
The importance of these pathways has made detailed biological information on their functions available. The interface location of the identified instances of the motifs raises the hope that they might be implicated in the dynamics of the information processing and, in particular, that the time-filter properties mentioned above might be exploited to control the time-response processing of the external signal. Ascertaining this behavior was our motive for investigating the detailed functioning of each of the pathways. We report here the principles of the core regulatory mechanisms involved in the chosen pathways, referring the reader to the cited literature for a detailed treatment. Here we are interested in identifying the possible role of motifs in biological functions.
The methionine pathway
Sub-network MET in Figures 2 and 4a shows the interaction graph for the cluster of interacting genes centered on CBF1, MET4 and MET28. The graph includes three motifs of type II.2, five of type III.5 and one of type III.7 (see Figure 1 for motif types). The methionine biosynthesis network has been thoroughly investigated [22–25] and a detailed biological model of the pathway is now available. Cbf1p, Met4p and Met28p form a heterotrimer that activates target genes of the sulfur pathway (MET genes). Inside the complex, only Met4p has direct transcriptional action, with Cbf1p being involved in chromatin rearrangement and Met28p tethering the complex to the DNA. The MET genes are activated by the complex, but are repressed when one of the final metabolites of the pathway, AdoMet, increases. Two loops drive the dynamics of complex availability, sketched in Figure 4a. One is a positive loop: the Met4p complex regulates the transcription of MET28, its product stimulating the tethering of the complex to DNA. This loop is responsible for the increase of the dynamic response when the intracellular AdoMet concentration is low (the transcription of MET4 is constitutive). The other is a negative loop: Met4p controls its own fate by regulating the transcription of MET30. The product of the latter is an ubiquitin ligase, which triggers the degradation of Met4p when AdoMet increases. This loop is expected to control high detrimental accumulation of AdoMet.
Note that the latter post-transcriptional mechanism is, by definition, not captured by the network, which is limited to transcriptional regulations. Furthermore, an intrinsic limitation of network structures should be noted: the three proteins Cbf1p, Met4p and Met28p always act as a complex. This information does not unambiguously emerge from the topology of the network (Figure 4a, left), as the topology is also compatible with the three proteins acting separately. In conclusion, the key features of the methionine synthesis pathway do not seem to hinge on transcriptional regulation via the motifs instances shown in Figure 4a.
Nitrogen catabolite repression (NCR) system
The NCR system shown in Figures 2 and 4b is used by the cell to control the synthesis of proteins capable of handling poor sources of nitrogen. NCR-sensitive genes are not activated when rich sources are available, whereas they get expressed when only poor sources are left. Two II.1 and one II.4 motifs are embedded in this system.
DEH1 and DAL80 are part of the GATA gene family and are known transcriptional repressors, regulating nitrogen catabolite repression via their binding to the GATA sequences upstream of NCR-sensitive genes. For several targets, the two repressors are in competition with Gln3p and Gat1p, which are transcriptional activators binding the same sequences.
The accepted mechanisms of NCR are as follows ([26–28] and see Figure 4b). First, in the presence of rich nitrogen sources (ammonia and/or glutamine), Gln3p and Gat1p are sequestered in the cytoplasm and can activate neither NCR-sensitive genes nor DEH1 and DAL80. The consequence of the low concentration of Gln3p in the nucleus is a low-level expression of DEH1, DAL80 and NCR-sensitive genes. Second, when poor sources only are available (such as urea, prolin, or GABA), Gln3p and Gat1p are released into the nucleus. The former activates GAT1 and the two proteins together activate NCR-sensitive genes. After a delay (due to the time taken for transcription and translation), Dal80p and Deh1p are expressed and competitively inhibit these same genes.
Interesting dynamic behavior takes place during a transition from rich to poor nitrogen sources, when the cell must cast about for alternative sources, which implies the synthesis of new proteins. The amount of these proteins synthesized must be sufficient to ensure utilization of the new sources but, because of the depletion of nutrient sources, they should not be too high. NCR-sensitive genes are therefore activated only for the limited period of time when Gln3p and Gat1p are present but Dal80p and Deh1p are not. The negative feedback of DAL80 on its activator GAT1 is the mechanism ensuring that oscillatory behavior.
To summarize, the role of the motifs identified in the NCR system is not evident and the entire mechanism of the NCR, within the model currently accepted on the basis of the present knowledge, can be described without any reference to them.
Pseudohyphal growth/mating MAPK system
The sub-network HYPHE in Figure 2 and Figure 4c is formed by one motif of type III.5, involving the two genes STE12 and TEC1. These genes both code for a transcription factor and are located downstream of the mitogen-activated protein kinase (MAPK) signal transduction pathway that controls both the pseudohyphal growth of the yeast and its mating response to pheromones. These signal transductions constitute a striking example of a signaling pathway shared by two different signals and yet responding specifically to each of them. It is therefore the object of detailed investigation and much data are available . The phenomenology of the regulatory process is summarized as follows: in response to pheromones, Ste12p binds specifically to the pheromone response elements (PRE) of genes involved in the mating process; under conditions of starvation, a heterodimer composed of Tec1p and Ste12p binds to genes involved in pseudohyphal growth.
The fact that STE12 regulates TEC1 raises the possibility that the switch between the two shared pathways of response to pheromones and pseudohyphal growth be realized by the instance of the feed-forward III.5 motif in the HYPHE sub-network. However, there is quite clear evidence that this is not the case, the most direct indication being provided in , where it is shown that the level of expression of TEC1 does not correlate with pseudohyphal growth. Recent work indicates that the switch is instead realized via post-transcriptional phosphorylation effects, controlled by the two kinases Fus3p and Kss1p, and affecting the multimerization of Ste12p. Fus3p and Kss1p constitute the final layer of the MAPK system and are differentially activated in the two pathways (see, for example ).
Regulation of early meiotic genes
The sub-network around IME1 in Figure 2 and Figure 4d is made of one II.1, two III.5 and one III.6 motifs and is implicated in the activation of early meiotic genes. The process of regulation of entry into meiosis and the early activation of the relevant genes has been studied in great detail and is summarized in . In short, the meiotic pathway in yeast is initiated by the expression and activation of IME1, which serves as the master regulatory switch for meiosis . Expression of IME1 requires the integration of a genetic signal, indicating that the cell is diploid, and a nutritional signal, indicating that the cell is starved. The point of interest here is to ascertain if the processing of these signals takes place at the transcriptional level by the instances of the motifs in the sub-network. This does not seem to be the case. The information processing is rather implemented by alternative routes and the picture of the interactions shown on the sub-network CCYCLE in Figure 2 and Figure 4d (left) appears to be insufficient and misleading.
The repression of IME1 by RME1 has a major role in cell-type control, and IME1 expression does not involve the regulation of RME1 by the complex Ume6p-Sin3p, as suggested by the sub-network CCYCLE in Figure 2. This is realized through the cell-type specific a1 and α 2 proteins, which combine in diploid cells and bind specifically to sites in the promoter of RME1 to repress its expression [32, 33].
The integration of the nutritional signal is processed by both IME1 and IME2 and is considerably more complex than cell-type regulation, its main steps being reviewed in . For instance, the IME1 promoter has at least 10 separate regulatory elements. IME2 is also regulated by several distinct signals, integrated at a single regulatory element, the upstream repression site URS1, which is bound by the Ume6p transcription factor under all conditions tested. The activation of IME1 and IME2 depends on the multimerization of Ume6p with several other proteins regulated either positively or negatively by at least two kinases, Rim11p and Rim15p. Other non-transcriptional mechanisms of gene control (such as targeted degradation) appear also to be involved in the regulation of this process . The motifs in the sub-network CCYCLE fail to capture the complexity of these interwoven interactions.
Pleiotropic drug resistance (PDR) system
The PDR system is used by the cell to counter the action of a broad spectrum of toxic substances; by activating membrane efflux pumps and modifying the membrane composition, the concentration of these substances is then decreased. Two genes, PDR1 and PDR3, encode homologous transcription factors [36, 37], which drive multidrug resistance by activating genes involved in active transport and lipid metabolism [38, 39].
The corresponding sub-network (named PDR in Figure 2 and 4e) is composed of eight motifs of type III.1 (so-called feed-forward loops) and one of type II.1, showing a star-like configuration with PDR1 and PDR3 in a central position.
In vivo, those two genes have apparent functional redundancy: they target the same genes and the deletion of either PDR1 or PDR3 does not significantly affect the PDR system; an effect is only shown when both are deleted [40, 41]. However, these two factors are used in response of two different cell signals: PDR3 is sensitive to mitochondrial activity, whereas PDR1 is not [42–44]. Conversely, PDR1 deletion mutants are quite drug-hypersensitive, whereas PDR3 mutants are not .
In addition to this distinct response of PDR1 and PDR3 to cellular signals, the regulation link between them is weak, and no proof of cooperativity for the regulation of their targets was highlighted.
It the PDR sub-network, the III.1 motifs formed by PDR1, PDR3 and their common targets are apparently not exploited by the cell because PDR1 and PDR3 are not obligatorily active at the same time and the prerequisites for the specific dynamics of feed-forward loops are not fulfilled (sufficient regulation of PDR3 by PDR1 and cooperativity on the common targets).
The motivating idea behind most discussions on motifs is the possibility of capturing the essential logic of genetic regulation by a small set of interaction circuits performing some specific functional tasks. While this hypothesis is, in principle, experimentally testable, experimental and theoretical work has hitherto considered essentially motifs in isolation, that is, excised from the biological environment in which the motifs' instances are embedded.
We studied in detail the role of motifs in the case of the best-documented genetic sub-networks and biological functions where such motifs are found. In most cases, motifs do not seem to have a central regulatory role in the biological processes associated with each occurrence. The list of examples where enough biological information is available is, of course, limited, and further examples may subvert this picture. At the moment, it is a fact that all the examples studied highlight the high level of integration of different regulatory mechanisms acting altogether. Reception and processing of cellular signals cannot be reduced to transcriptional regulation and protein-protein interaction switches. Other mechanisms such as phosphorylation, triggered degradation, protein sequestration and transport, and higher-order multimerization are central to the logic of the sub-networks. Disentangling information-processing circuits made of transcription reactions and interactions between transcription factors from the whole cellular environment does not seem to be possible for the cases considered. A qualitative impression surmised from the visible aggregation and nesting of the motifs with the rest of the network is that a 'pure' modular functional behavior is not very likely to occur. This impression is not limited to S. cerevisiae: in previous work , other researchers have shown that a similar aggregation of structural motifs occurs for a simpler organism, E. coli, suggesting some degree of generality.
Some comments on structuring interaction data in the form of topological networks are worth making. The graph is indeed an abstraction constructed from available databases and its meaning is influenced by several factors. For instance, the graph is a static projection of possible interactions. The analysis of regulatory processes varying in space and time requires additional information not usually included in the topology of biological networks. Indeed, the very representation in the form of a unique network entails the integration in space and time of the interactions taking place during the cellular lifetime. Some of the patterns of interaction might then be spuriously due to a projection effect, whereas they actually take place at different times and/or locations within the cell. This is occurring, for example, in the PDR system: PDR1 and PDR3 at the base of the eight III.1 motifs respond to different signals and control their outputs independently (no cooperation on the common targets). These motifs appear in the network because different conditions at different times were projected onto the same plane.
Furthermore, the patterns in the network may be a direct consequence of the data models in the current databases, and incorrectly represent the biological context. Transitory macromolecular associations like protein complexes and interactions between a whole protein complex and a target are indeed missed, and at most represented as individual links between each component and the target. This is what occurs with the Met4p/Met28p/Cbf1p heterotrimer, which appears in the network as three independent interacting components together with three III.5 motifs that do not actually exist.
The NCR system is an interesting example where motifs are clearly identified and seem unambiguous. However, to the best of our knowledge they do not play any significant role. In particular, the role of the mutual interactions between DAL80 and DEH1 (sustaining a II.4 motif) is not clear. An intriguing hypothesis is that the presence of the interactions might be traced back to the strong sequence similarity between DAL80 and DEH1. The products of both these genes form homodimers and inhibit their own expression. The presence of the motif might then be due to a recent duplication event, which has therefore preserved the interactions.
Divergent evolution seems also to be the origin of the appearance of motifs in the PDR system. In this case, the two diverging genes PDR1 and PDR3 have acquired different independent functions. The motif instance that they form together is the apparently unexploited consequence of their common origin.
The results presented here indicate that the statistical abundance of network motifs has no evident counterpart at the evolutionary and in vivo functional level. Occurrences of network motifs have indeed been shown to possess the same evolutionary fragility; that is, when different organisms are compared, the genes composing the motif have similar co-occurrence profiles as genes in interaction patterns with a normal abundance.
The point seems to be confirmed by the analysis of the functional role of examples of the motifs occurrences. These are located at the interface between two blocks - one responsible for the reception of a signal and the other for its processing - and have been selected because detailed biological information on those pathways is available. The number of cases is limited, but in none of them are the major steps of signal information processing taking place at the transcriptional level through the implementation of the motifs. Alternative routes involving post-transcriptional regulation and intracellular compartmentalization seem to be exploited for this purpose.
These results naturally bring up the issue as to the actual role of the motifs. Some occurrences have been shown to arise spuriously from the representation of the interaction data in the form of a network and the ensuing projection effects in space and/or time. It seems, however, fair to assume that those effects should be limited to a few cases. The metabolic costs of producing proteins and the fact that some of the motifs instances examined are active in conditions of starvation make it likely that proteins encoded by genes composing these motifs do play a role. What is however quite clear from Figure 2 and our analysis is that the great majority of motif occurrences are in fact embedded in larger structures and entangled with the rest of the network. Only a small minority is isolated and likely to perform a specific functional task that does not depend on the context.
This clustering is important as it indicates that the choice of the null model used to gauge the statistical importance of the abundance of interaction patterns might be delicate. Indeed, the higher-order context is not taken into account in the randomization process used to generate the null model networks, and we have shown that this is manifestly not a choice ensuring a strong evolutionary and (in vivo) functional significance. Accounting for the various layers of organization of biological networks seems crucial to correctly identify the functional elements responsible for the information processing that allows living cells to cope with their highly variable environmental conditions.
Materials and methods
The transcriptional regulatory network used for the analysis is the one constructed and investigated in . It was preferred to the more extended one derived from ChIP-chips data in  as the fraction of links where the regulatory role of the various interactions is documented is higher for the former. The protein-protein interaction data in the Database of Interacting Proteins (DIP ) are a large collection of both two-hybrid and TAP-tag data. The resulting network has 476 nodes, 905 directed transcriptional edges and 221 undirected protein-protein edges.
Identification of motifs and network randomization
The detection of n-node network motifs is performed along lines similar to those used in . The method exhaustively scans the neighborhood of all the links in the network to search for the motif of interest, and then purges the list for repeated patterns.
Randomized versions of the network are generated as follows. Links are swapped as in the Markov-chain algorithm used in , that is, two links between the couples of nodes (X1Y1) and (X2Y2) are replaced by (X1Y2) and (X2Y1). In our case, where the links might be transcriptional or protein-protein interaction, the links that are swapped must be of the same type. This procedure is guaranteed to preserve the single-point connectivity at each node of the network.
As for the randomization procedure for n = 3 motifs, we want to avoid the possibility that higher-order motifs spuriously inherit statistical significance from lower orders. In other words, the randomized network ought to have the same statistics for all the patterns of order n = 2 as the real network. This is ensured by converging a simulated annealing, where the elementary steps are the swappings of the links previously described. The transition probabilities are weighted according to the difference:
where the sum runs over all the patterns of order n = 2 and the c i values denote the number of patterns in the two types of networks.
- or the opposite inequality if the pattern is under-represented in the real network - to occur in the randomized ensemble. The probabilities are estimated from a Monte-Carlo sampling of 10,000 trials of the randomized ensemble distribution and the results are sensitive neither to the number of trials nor to the thresholds chosen. The probability distribution functions are often found to deviate from a Gaussian curve and the one-tailed probabilities are therefore directly measured from the normalized histograms without relying on z-scores.
Note that patterns involving self-interactions are somewhat special, as their order n, which controls the type of random networks they should be compared to, does not coincide with their number of genes. For example, a single gene self-interacting is treated as an n = 2 pattern. The reason is that a sensible way of assessing the significance for this pattern is by having a fixed number of total proteic links and studying the fraction of them that are self-interactions. In other words, self-interactions are swapped throughout the randomization procedure with proteic links between two distinct proteins and their order is therefore n = 2.
BLAST searches were performed using BLASTP 2.2.6  with the BLOSUM 62 matrix and affine gap penalties of 11 (gap) and 1 (extension). Putative orthologs were inferred from the primary sequence and keeping only bidirectional best hits to reduce the effect of the high number of paralogs in yeast genomes. Tables of bidirectional best hits were constructed by identifying the pairs of proteins in the two organisms compared which are the reciprocal best alignments. The significance of the alignments was quantified by the BLAST e-values and different thresholds were considered, ranging from 10-1 to 10-10. Their choice does not affect the results presented in the body of the paper.
Evolutionary fragility of interaction patterns
Let us consider all the interaction patterns, indexed by i, composed of interacting genes of S. cerevisiae and each one of the other four hemiascomycetes, indexed by α. The boolean variable f iα for the pattern i is taken equal to zero if the genes composing the pattern are all present/absent in the other organism α and is unity otherwise. Presence/absence is measured by using the list of bidirectional best hits discussed in the previous section. The selective pressure to preserve the pattern i is quantified by the fragility:
The two extreme cases are F i = 0 and F i = 4 (the number of organisms compared). The two cases correspond to the genes composing the pattern co-occurring in all or none of the compared organisms, respectively. As an additional example, consider the case where the three genes composing an interaction pattern are all present in C. glabrata, K. lactis and D. hansenii (which are evolutionarily closer to S. cerevisiae) but one (or two) of them is absent in Y. lipolytica. The corresponding value of the fragility is F i = 1.
Additional data files
We are grateful to B. Dujon, P. Glaser and F. Képès for useful discussions. M.V.'s research was supported in part by the National Science Foundation under Grant No PHY99-07949.
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