Monotonicity, frustration, and ordered response: an analysis of the energy landscape of perturbed large-scale biological networks
- Giovanni Iacono^{1} and
- Claudio Altafini^{1}Email author
https://doi.org/10.1186/1752-0509-4-83
© Iacono and Altafini; licensee BioMed Central Ltd. 2010
Received: 24 November 2009
Accepted: 10 June 2010
Published: 10 June 2010
Abstract
Background
For large-scale biological networks represented as signed graphs, the index of frustration measures how far a network is from a monotone system, i.e., how incoherently the system responds to perturbations.
Results
In this paper we find that the frustration is systematically lower in transcriptional networks (modeled at functional level) than in signaling and metabolic networks (modeled at stoichiometric level). A possible interpretation of this result is in terms of energetic cost of an interaction: an erroneous or contradictory transcriptional action costs much more than a signaling/metabolic error, and therefore must be avoided as much as possible. Averaging over all possible perturbations, however, we also find that unlike for transcriptional networks, in the signaling/metabolic networks the probability of finding the system in its least frustrated configuration tends to be high also in correspondence of a moderate energetic regime, meaning that, in spite of the higher frustration, these networks can achieve a globally ordered response to perturbations even for moderate values of the strength of the interactions. Furthermore, an analysis of the energy landscape shows that signaling and metabolic networks lack energetic barriers around their global optima, a property also favouring global order.
Conclusion
In conclusion, transcriptional and signaling/metabolic networks appear to have systematic differences in both the index of frustration and the transition to global order. These differences are interpretable in terms of the different functions of the various classes of networks.
Keywords
Background
From a statistical physics perspective, the problem of determining monotonicity (or near monotonicity) is equivalent to checking when an Ising model with signed interactions has no (or little) frustration [21, 23]. In terms of the signed graph, frustration corresponds to undirected cycles having an odd number of negative edges [21]. See also [27] for another recent use of Ising models in the context of complex networks. In this work we are interested in computing the frustration of biological networks of various types: transcriptional, signaling and metabolic. When modeling these different classes of networks as signed graphs, we have to use different levels of resolution: for signaling and metabolic networks we start from a set of stoichiometric reactions and obtain the signed graph by taking the signature of the Jacobian of the corresponding reaction kinetics, hence an edge represents the contribution of a molecular specie to a kinetic reaction, see [8, 23, 26] and the Methods Section. For transcriptional networks, on the contrary, we model interactions at functional level, i.e., we take an edge to represent the entire action of activation/inhibition of a transcription factor on a target gene, and in doing so we lump together many important molecular steps, from the binding of the transcription factor to the promoter region of a target gene to the final release of the newly synthetized mRNA molecule. Energetically, such complex process is various (or many) times more relevant than a signaling event or a metabolic reaction. Also the corresponding time scales differ by several orders of magnitude [4, 28]. Of course, we are forced to use this coarser level of resolution because the stoichiometric details are different for different transcriptional actions, and are not known systematically (see [29] for the only example we know of in this direction). Notice that a similar functional representation, oriented at capturing the "information flow" rather than the "mass flow", is possible also for signaling networks [4, 7, 9, 10]. Although it may elucidate better the causal transfer of "information" along the pathways, it seems less appropriate to describe the energetic content of the biochemical transformations necessary for the propagation of the signal than the stoichiometric level which we use in this paper, see Supplementary Notes in Additional File 1 for a more detailed discussion. In any case, the qualitative difference in the modeling assumptions made should always be kept in mind, and the classes of networks analyzed should be connotated accordingly as "transcriptional, at functional level" and "signaling/metabolic, at stoichiometric level".
Under these assumptions, the frustration index we observe varies considerably according to the type of network analyzed: it is very low for gene regulatory, networks and much higher for signaling and metabolic networks. In this paper we propose an interpretation of this different behavior based on the characteristic "energy" associated to the interactions of a graph. We assume that the costs of the interactions (i.e., the weights of the edges) are all comparable on each class of networks, but not across classes of networks. In particular, transcriptional edges have a much higher cost than the other classes of interactions, and we can speculate that on an evolutionary scale this may have strongly disfavored the development of interactions leading to frustration, i.e., of incoherent or contradictory transcriptional orders. For the "cheaper" signaling and metabolic interactions, instead, such a tight control may not be required, especially since a higher frustration may induce a richer and more complex dynamical behavior.
We know from the theory of Ising models that it is energetically favorable for neighbouring spins to be aligned when the interaction constant is positive and to be antialigned when it is negative. If we associate to the frustration index the global optimum of an "energy" function describing the amount of such unsatisfied interactions, then we can say that networks with low frustration will have a "ground state" (i.e., a global optimum) of lower energy than more frustrated networks. In addition, rather than just focusing on the energy of the optimal configuration, we can average the state of the system over all possible perturbations, and study what is the average frustration of a network. In particular, then, if we take the strength of the interactions of a network as "cooling" parameter, we can use statistical physics arguments [15] to describe how the probability of occupancy of the global minimum of the energy varies with the interaction strength, and therefore how monotonically a network behaves in average in response to random perturbations. What we observe is that the more frustrated signaling/metabolic networks achieve "order" (i.e, tend to populate their global minimum of energy) in a range of interaction energies which is lower than for the transcriptional networks, meaning that these networks (in average) tend to respond to perturbations as coherently as they can even for moderate values of energy. This behavior partially compensates for the higher frustration, which, as already mentioned, might be instrumental to the achievement of more complex dynamics than those required for the transcriptional networks. The transcriptional networks, on the other hand, only contain strong interactions and are therefore not concerned with the lower energetic regime. Coherently, they show a topological structure richer in tree-like subgraphs, which disfavor the transition to ordered behavior, and which are absent in the other classes of networks.
That signaling and metabolic networks may require a lower energetic content to experience a transition to ordered behavior is also confirmed by the structure of their energy landscapes which, unlike for the transcriptional networks, lack high and neat energetic barriers around the global optima, meaning that reconfiguration to the ground state can be easily achieved even at modest energies.
Results
which expresses the total cost associated to the perturbation s. Assuming that all interactions of a network have the same strength, |J_{ ij }| = 1 whenever J_{ ij }≠ 0, the cost of each interaction depends on the sign of each nonzero J_{ ij }: for J_{ ij }> 0 (activator) the aligned s_{ i }, s_{ j }spin configuration is more energetically favorable (-J_{ ij }s_{ i }s_{ j }= -1 < 0) than the antialigned one (-J_{ij}s_{ i }s_{ j }= 1 > 0) and viceversa for J_{ ij }< 0. Of all 2^{ n }possible spin assignments, those respecting monotonicity will be such that J_{ ij }s_{ i }s_{ i }> 0 on each edge of the graph, i.e., those contributing to minimizing h(s). A spin system is said frustrated when not all these constraints J_{ ij }s_{ i }s_{ j }> 0 can be satisfied simultaneously by any assignment. Computing how far a given network is from being monotone corresponds to computing the ground state s_{ground}, i.e., the spin assignment that globally minimizes (1). It has been shown [23] that this is an NP-hard problem, equivalent to the MAX-CUT problem or, in terms of the Ising model, to computing the exact frustration index of the network [21, 30], call it δ. In [26] (see also Supplementary Notes in Additional File 1 for a quick recap), we proposed efficient heuristic algorithms providing fairly tight upper and lower bounds for δ in biological networks of the size of the thousands nodes. From the theory of monotone systems (see [21, 22] and the Methods), is monotone if and only if there exists a diagonal signature matrix D_{ σ }(i.e., a matrix having on the diagonal the vector σ of elements σ_{ i }∈ {±1}) such that has all nonnegative entries, see Lemma 2.1 in [22]. _{ σ }and have different sign patterns but the same frustration index δ, as D_{ σ }is a change of sign through a cut set of the graph of and such "gauge transformations" D_{ σ }[31] leave the sign of each cycle of the graph (and hence δ) unaltered.
Let us consider first as an illustrative example the yeast cell cycle network introduced in [19] in the context of boolean networks, see Fig. 1. With respect to the original graph of [19], we drop the self-loops and consider the underlying undirected graph (only a pair of edges is incompatible with this symmetrization of the adjacency matrix). The number of negative signs on the symmetrized adjacency matrix is 10. However, a gauge transformation on the three nodes Cib1,2 Clb5,6 and Cln1,2 yields a _{ σ }with only 4 negative edges, which is a global optimum for the frustration index δ, see Fig. 1(a) . The presence of frustrated cycles in a network leads to a lack of coherence in the response of the system to perturbations.
This can be observed in Fig. 1(b), where the response of the yeast cell cycle and of a monotone network built on the same graph are compared. The behavior of the non-monotone cell cycle network is less predictable and potentially contradictory (see also Fig. S2 for analogous considerations on the simpler feedforward loop example [5]). It is then important to have an estimate of how close a true network is to being monotone i.e., frustration-free. Our algorithms allow to obtain a _{ σ }with as low as possible number of negative signs also for large-scale networks. This number is typically close to δ, meaning that it is now much easier to localize on the graph of _{ σ }the potentially frustrated edges (or, more properly, the frustrated cycles). Another consequence is that the candidate ground state for _{ σ }that globally minimizes (1) is now straightforward to identify, as it corresponds to the "all spins up" configuration, call it 1. Hence, the candidate ground state for the original can be found reversing the gauge transformation: s_{ ground }= D_{ σ }1. Approximate values for the frustration index δ and for the corresponding energy minimum not very far from the true ones can therefore be computed.
Frustration in large-scale biological networks
Networks used in this study.
Network | n | m | leaves | description | ||
---|---|---|---|---|---|---|
transcriptional | level of detail: functional | |||||
E.coli | 1475 | 3320 | 556 | gene regulatory network of the E.coli, from RegulonDB database, ([42], http://regulondb.ccg.unam.mx), version 6.3. | ||
Yeast | 690 | 1082 | 348 | gene regulatory network of S.cerevisiae, from [5] | ||
B.subtilis | 918 | 1324 | 528 | gene regulatory network for Bacillus Subtilis, assembled by [43] | ||
Cory | 344 | 366 | 264 | Corynebacteria gene regulatory network (experimental interactions only). Assembled by [44] | ||
signaling | level of detail: stoichiometric | |||||
EGRF | 330 | 852 | 12 | Epidermal Growth Factor Receptor pathway. Created by [45] | ||
Toll-like | 679 | 2204 | 59 | Signaling network for the Toll-like-receptor. Assembled by [46] | ||
metabolic | level of detail: stoichiometric | |||||
E.coli | 757 | 6116 | 84 | metabolic network of E.coli, from [47] | ||
Yeast | 797 | 4436 | 17 | metabolic network of the yeast S.cerevisiae. Assembled from [48] |
Data for the frustration index δ.
Network | δ_{ low } | δ_{ up } | δ_{ max } | δ_{ null } | σ_{ null } | Z_{ score } | Pvalue |
---|---|---|---|---|---|---|---|
transcriptional | |||||||
E. Coli | 365 | 371 | 1579 | 662,86 | 9,77 | 29,86 | p≪ 10^{-100} |
Yeast | 41 | 41 | 401 | 116,67 | 5,83 | 12,98 | p = 8 · 10^{-39} |
B. Subtilis | 71 | 71 | 415 | 139,73 | 6,53 | 10,52 | p = 3,5 · 10^{-26} |
Cory | 9 | 9 | 48 | 71,15 | 2,16 | 3,76 | p = 8,3 · 10^{-5} |
signaling | |||||||
EGFR | 183 | 193 | 375 | 149,75 | 5,01 | -8,62 | p = 3,3 · 10^{-18} |
Toll-like | 401 | 468 | 873 | 384,92 | 7,70 | -10,78 | p = 2,1 · 10^{-27} |
metabolic | |||||||
Yeast metab | 670 | 747 | 1421 | 667,42 | 10,3 | -7,72 | p = 5,6 · 10^{-15} |
Ecoli metab | 912 | 1017 | 1944 | 1006,9 | 12,73 | -0,79 | p = 0,21 |
Average frustration and ordered response
The values of δ and h(s_{ ground }) alone are not enough to characterize how monotonically the system behaves in average. In fact, the energy landscape of frustrated Ising spin systems is known to be usually rugged [37, 38], and the presence of a single deep minimum in (1) is not enough to guarantee that the energy averaged over all configurations s (corresponding to all possible multinode perturbations) is indeed more negative than in other systems whose energy landscape is characterized by valleys which are maybe less deep but with larger basins. In other words, to characterize how monotone is the response of the system to arbitrarily complex perturbations we have to consider the average value that h(s) assumes over all possible spin assignments, weighted by the probability of each s. This "internal energy", call it ⟨h⟩, is an indicator of how coherently the system is behaving in average: the more negative ⟨h⟩ is, the less the responses of the system to perturbations are "contradictory" at some fan-in node or along directed cycles. Denote with the partition function of the system, β ∈ ℝ_{+}. As usual in statistical physics, the partition function Z is the normalization factor that renders the frequencies of the various spin states true probability densities. For spin systems, β has the meaning of an inverse temperature and it is normally used as "cooling" parameter, i.e., when β→ ∞ the probability of the state s, p(s), tends to concentrate on the ground states: p(s_{ ground }) → 1 as β → ∞. In the context of biological networks, the temperature is taken as ~ 298 K and it is not a varying parameter. However, we can use β to describe the strength of the interactions of a network. Recall that in forming the energy (1), was taken as a signed adjacency matrix with all interactions equal to 1, regardless of the nature of the network studied. As a matter of fact, metabolic, signaling and transcriptional interactions are characterized by widely different energetic costs. In particular, if in our stoichiometric representation a metabolic reaction or a signaling event might have a comparable energetic content, a link in a gene regulatory network describes the entire cascade of events in which the transcription of a gene can be broken down and overall its cost is much higher than in the other networks. Hence, in our fixed temperature context, taking into account the interaction cost β rescales h(s) to the "absolute" energy βh(s). The probability of a given configuration s, p(s) = e^{-βh(s)}Z(β)^{-1}, is a function of β and is maximized in the (usually degenerate) ground state s_{ ground }. As for spin systems, , i.e., when β is large enough, in average the system will always be found in the configuration s_{ ground }which minimizes the energy (1) and which exhibits the least frustration for the network.
The qualitative difference in the phase transition to order between transcriptional and signaling/metabolic networks suggests an interpretation coherent with the different energetic content associated to the classes of networks. In fact, we can say that since β_{ transcr }is high, it is much less plausible for a transcriptional network to be operating in a regimen of low β than it is for signaling/metabolic networks. On the contrary, for these last two classes of networks, it is not unlikely to have interactions of medium-low strength. Hence it gets much more important that ⟨s_{σ}⟩ → 1 even in correspondence of moderate values of β, because this helps in maintaining a coherent behavior in response to perturbations, as required in order to carry out correctly a biological task.
Sampling the energy landscape
Discussion
For a gene regulatory network, an edge represents the cost of the entire action of transcription of a gene. This is a complex, multistep process: for prokaryotes, for example, it includes the binding of the transcription factor to the DNA, the recruitment of a polymerase, the unwinding of the DNA helix, the detachment of the σ -factor and the conformational changes in the polymerase preceeding elongation, the release of both the DNA and of the complete mRNA at the termination phase. The energetic cost and time constant of such a complex process are relevant for a cell. Hence, especially in lower organisms, it is natural to expect that in a transcriptional network the genes behave in concert and that the fraction of the gene-gene interactions that contribute to minimizing the energy in response to perturbations is substantially larger than in a metabolic or signaling network, as a frustrated bond costs much more to the cell and its effect lasts much longer. In particular, frustrations manifest themselves on the cycles of the underlying undirected graph of the network as contradictory transcriptional orders. While changing the transcriptional commands is necessary to cope with e.g. different environmental conditions, encoding them as frustrated cycles can easily lead to unpredictable or erroneous dynamical behavior. Therefore, in spite of the presence of certain characteristic motifs leading to frustration (the incoherent feedforward loops mentioned in [5, 6] for the E.coli and Yeast transcriptional networks are common examples), overall the transcriptional networks we analyze are indeed near-monotone. Both the topology and the sign assignments to the nodes of the transcriptional networks contribute to achieve a degree of monotonicity which is higher than expected from null models. On the contrary, incoherent signaling or metabolic actions are energetically much less relevant than a single transcriptional event and can be easily tolerated by the cell, especially since nonmonotone patterns favour a richer dynamical behavior. While the level of detail at which we model our networks (functional for transcriptional networks, stoichiometric for signaling and metabolic networks) certainly contributes to the systematic differences in the frustration index, other factors such as the tendency of the transcriptional networks to have skewed sign distributions are also crucial in attaining a low frustration. It is interesting, then, to notice that in E.coli the transcription factors violating this rule are primarily involved in the mediation of external signaling, rather than in regulatory or structural functions (Table S5).
For spin systems, the tendency to satisfy pairwise all interactions grows when the temperature decreases, although in a frustrated Ising spin system all the conditions can never be satisfied simultaneously. In this paper, we consider the strength of the interactions as the key factor that determines the increase in the probability of finding the system in its ground state (i.e., in its least frustrated/maximally monotone configuration). If we parametrize the networks by the interaction strength and study the probability of finding the system response in the ground state as a function of this cost, we observe that for signaling/metabolic networks it is higher than for transcriptional networks in the region of medium/low values of the interactions. This behavior, which is due to the topological structure of the networks and to the energy landscape it determines, could reflect the tendency of signaling/metabolic networks to attain a globally ordered response in spite of the weaker energetic content of their interactions. As such, it helps maintaining coherence of the response in spite of the higher level of frustration of these networks (which, again, favors a richer dynamical behavior). For transcriptional networks, on the other hand, owing to the strong interactions, the regime of low energies is less important, hence tree-like motifs, which hinder the establishment of long-range correlations, are abundant.
A Montecarlo investigation of the energy landscape of the networks [37, 41] suggests that transcriptional networks tend to have a more funneled landscape than the other networks (at least around the global optima), with a single deep well of global minima delimited by high barriers, while in signaling and metabolic networks the optima are surrounded by local minima of comparable energy. Order in these classes of networks is favored also by the lack of neat energetic barriers separating local and global optima, which enables the reconfiguration to the global optimum through low-energy paths.
Several are the caveat and limitations of our study. First of all, the different levels of resolution for the different classes of networks may be a source (or the source) of the systematic differences we are observing. Hints in this direction come for example from the observation that networks at functional level tend to have less cycles than networks at stoichiometric level (see Supplementary Notes in Additional File 1 for the origin of this fact), and that functional models of signaling pathways may also have asymmetric sign distributions (for example non-specific kinases catalyzing the phosphorylation of various proteins will have many positive edges, while non-specific phosphatases will have multiple negative edges). This is observed to some extent in the functional model of the hippocampal signaling network proposed in [9]. Notice that this network has a large fraction (approximately a third) of interactions representing protein-protein or protein-ligand bindings, to which it is unclear how to associate a sign in an unambiguous manner. The ambiguity of course also propagates to the level of frustration one obtains correspondingly. More generally, we are not aware of any systematic way to map the pathway charts available at stoichiometric level to the functional level, allowing to univocally assign a sign to each edge without at the same time loosing in this process a large part of the molecular species involved. Notice also that the opposite option, namely representing transcriptional networks at stoichiometric level, is de facto impossible with our current knowledge.
Another important source of uncertainty comes from the limited coverage of the biological networks currently available. In particular, for the transcriptional networks, the fraction of target genes having at least a transcription factor is below 50% of the genes. Furthermore, our considerations about an higher than expected monotonicity may very well be overturned once more complex organisms (for whom the regulatory mechanisms are expected to be much more complex) are taken into account.
Conclusion
In conclusion, we have observed that distinct classes of biological networks seem to be characterizable by different features in response to perturbations. At least when we model transcriptional networks at functional level (i.e., as activation/inhibition links) and signaling and metabolic networks at stoichiometric level, we can observe that transcriptional networks appear to be less frustrated than expected and much less frustrated than signaling and metabolic networks, meaning that they might admit highly coherent responses to perturbations. On the other hand, the signaling/metabolic networks seem to have the ability to achieve an average ordered response in a lower range of interaction strengths than the transcriptional networks. We explain the first feature as the need to avoid as much as possible erroneous or contradictory transcriptional actions which would cost much more to the cell than analogous incoherent signaling/metabolic events. The second feature may partially compensate for the higher frustration of these last networks, by lowering the interaction strength needed for a transition to ordered response (in average), and thereby ensuring the effectiveness of this reduced coherent behavior in an energetic range more critical for these classes of networks.
Methods
i.e., _{ ns }of elements J_{ ns },_{ ij } ∈ {±1,0} is the signed adjacency matrix of a directed graph representing our network. Coherently with J_{ ns },_{ ij } ∈ {±1,0}, also the magnitude of the perturbations z is to be considered as unknown except for its sign: s = sign(z), meaning s_{ i }∈ {±1} ∀i = 1,... n. The entries J_{ ns },_{ ij }of the matrix _{ ns }represent the effect of the j-th variable on the i-th variable which can be activatory, J_{ ns },_{ ij }> 0, inhibitory, J_{ ns },_{ ij }< 0, or inexisting, J_{ ns },_{ ij }= 0. In general, this effect can change of sign with the operating point x_{o}[21], but we shall not consider this scenario here. As a matter of fact, it is worth remarking that for common choices of f(x), such as mass-action or Michaelis-Menten, the partial derivatives have indeed constant sign .
If, rather than in the directed graph of adjacency matrix _{ ns }, we are interested in the underlying undirected graph (resulting by dropping the arrows in the edges), then this is obtained symmetrizing the matrix _{ ns }Denote such symmetric signed adjacency matrix. The symmetrization operation is always possible as long as edge pairs J_{ ns },_{ ij }and J_{ ns },_{ ji }are compatible, i.e., J_{ ns },_{ ij }J_{ ns },_{ ji }≥ 0. In all of our networks, the symmetrization operation leads to very few or no conflicting signs at all, see Table S1.
Monotone dynamical system
As explained in detail in [21], the non strict inequality for monotonicity allows to test such conditions (3), rather than in the original directed graph of (2), on its underlying undirected counterpart, in which we conventionally drop the self-loops (for which σ_{ i }σ_{ i }J_{ ns,ii }> 0 if and only if J_{ ns,ii }> 0, i.e., the order relations (3) are trivial). Therefore, from now on we shall consider only the symmetrized version of _{ ns },with all diagonal elements fixed to 0, i.e., the matrix . Practically, this symmetrization operation means that we are interested not only to "true" directed cycles and their frustration, but also to multiple directed paths starting and ending on the same nodes (and forming cycles on the underlying undirected graph). See the feedforward examples in Supplementary Notes in Additional File 1 and Fig. S2.
Mean field approximation in heterogeneous signed networks
As , the energy is invariant to the gauge transformation D_{σ}. In fact, from s_{σ} = D_{σ}s, we have . Hence the mean field calculations for _{σ} are valid also in the original . In addition, however, as s_{σ,ground}= 1, in the gauge transformed system we have that, as β → ∞, ⟨s_{σ}⟩ → 1, a property which is in general not verified in the original basis, which will instead concentrate at its own ground state . Therefore, for all practical purposes, ⟨s_{σ}⟩ can be taken as "order parameter" of the spin glass. In fact, since the gauge transformation minimizes the number of negative edges, it also maximizes the number of spins whose value is +1 in the ground state. Hence, just like in a ferromagnet (i.e., in a spin system in which for all nonzero J_{ ij }one has J_{ ij }= +1), the average value of s_{σ} (i.e., the "magnetization") tends to 1 when the system is "cooled".
Declarations
Acknowledgements
The authors would like to thank K. Nakai for providing the transcriptional network of B.subtilis. This work was sponsored in part by a grant from Illy Caffé, Trieste, Italy.
Authors’ Affiliations
References
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