Evolvability of feed-forward loop architecture biases its abundance in transcription networks
- Stefanie Widder^{1, 2}Email author,
- Ricard Solé^{2, 3} and
- Javier Macía^{2}
DOI: 10.1186/1752-0509-6-7
© Widder et al; licensee BioMed Central Ltd. 2012
Received: 11 October 2011
Accepted: 19 January 2012
Published: 19 January 2012
Abstract
Background
Transcription networks define the core of the regulatory machinery of cellular life and are largely responsible for information processing and decision making. At the small scale, interaction motifs have been characterized based on their abundance and some seemingly general patterns have been described. In particular, the abundance of different feed-forward loop motifs in gene regulatory networks displays systematic biases towards some particular topologies, which are much more common than others. The causative process of this pattern is still matter of debate.
Results
We analyzed the entire motif-function landscape of the feed-forward loop using the formalism developed in a previous work. We evaluated the probabilities to implement possible functions for each motif and found that the kurtosis of these distributions correlate well with the natural abundance pattern. Kurtosis is a standard measure for the peakedness of probability distributions. Furthermore, we examined the functional robustness of the motifs facing mutational pressure in silico and observed that the abundance pattern is biased by the degree of their evolvability.
Conclusions
The natural abundance pattern of the feed-forward loop can be reconstructed concerning its intrinsic plasticity. Intrinsic plasticity is associated to each motif in terms of its capacity of implementing a repertoire of possible functions and it is directly linked to the motif's evolvability. Since evolvability is defined as the potential phenotypic variation of the motif upon mutation, the link plausibly explains the abundance pattern.
Background
which is usually dubbed as the genotype-phenotype mapping problem [4]. Understanding the nature and origins of this mapping is at the core of many key questions concerning the evolution of complexity in nature.
Within the context of gene transcription networks, it has been suggested that the previous problem can be dissected by analyzing the frequency of some overabundant sub-networks of three or four elements, so called network motifs [5–7]. These sub-graphs only capture the topological pattern of connections and a dynamical description of their potential function requires a set of differential equations [8, 9]. One particularly important example is provided by feed-forward loop (FFL) motifs [5]. Many genetic and biochemical systems, such as the Lac and Che systems in E. coli (responsible for lactose utilization and chemotaxis, respectively) involve FFL motifs [10–12]. Mounting evidence indicates that they have key roles in cell function [10] and morphogenesis [13, 14].
However, the origin of a preferential bias towards given topologies remains under discussion.
Results and Discussion
Probability distribution of implementing different functions
Consider the FFL graphs Γ_{ i }from the set $\mathcal{S}=\left\{{C}_{1},{C}_{2},{C}_{3},{C}_{4},{I}_{1},{I}_{2},{I}_{3},{I}_{4}\right\}$ shown in figure 1a. In previous studies [9, 26] it has been shown that, the topology of a given FFL does not univocally define its function but it captures the probability distribution of implementing different functions. Our first goal is to identify an appropriate mapping f between FFL topology and each potential response ϕ_{ j }(t), i.e. f_{ ij }: {Γ_{ i }, x(0), μ} → ϕ_{ j }(t) where we indicate as {Γ_{ i }, x(0),μ} the FFL graph together with its initial condition x(0) and the set of parameters used μ [26]. The six different responses (figure 1b) are triggered by an external input. These are either fast response (pulser) or delayed response (grader) considering the target concentration of the output. Here f_{ ij }indicates the likelihood that ϕ_{ j }(t) is implemented by Γ_{ i }.
Kurtosis is the measure of the "peakedness" of a distribution. It quantifies the concentration of frequencies around the mean of the distribution. Higher kurtosis means that the variance is the result of infrequent, extreme deviations from the mean as opposed to frequent, modestly sized deviations resulting in low kurtosis. In order to define a measure characterizing the degree of plasticity of a given motif in terms of its specialization or its flexibility, we can consider two extreme cases, namely the most flexible graph Γ_{ f }equally likely to implement any function ϕ_{ j }, and the most specialized graph Γ_{ s }implementing only one function ϕ_{1}. In the first case we would have P(ϕ_{ j }|Γ_{ f }) = 1/6 and the kurtosis associated is K(Γ_{ f }) = -3.33, whereas in the second case we would have P(Γ_{ s }) = 1 and 0 otherwise, with kurtosis K(Γ_{ s }) = 6. Details on the calculation can be found in Methods. Any other FFL graph Γ_{ i }from the set $\mathcal{S}=\left\{{C}_{1},{C}_{2},{C}_{3},{C}_{4},{I}_{1},{I}_{2},{I}_{3},{I}_{4}\right\}$ has kurtosis values locating within this interval, K(Γ_{ i }) ∈ (K(Γ_{ f }), K(Γ_{ s })).
where α is a normalization coefficient defined as $\alpha ={\sum}_{j=1}^{8}{\psi \left({\Gamma}_{j}\right)}^{-1}$.
Kurtosis, ψ(Γ_{ i }) and predicted probability ρ(Γ_{ i })
Kurtosis | ψ(Γ_{ i }) | ρ(Γ_{ i }) | |
---|---|---|---|
C1 | 1.631 | 0.297 | 0.342 |
C2 | 4.142 | 2.808 | 0.036 |
C3 | 3.042 | 1.707 | 0.059 |
C4 | 4.835 | 3.501 | 0.029 |
I1 | -1.721 | 0.386 | 0.263 |
I2 | 2.506 | 1.17 | 0.087 |
I3 | -2.083 | 0.748 | 0.136 |
I4 | 3.459 | 2.123 | 0.048 |
For the less abundant motifs we see a more disordered trend in the two measures, as is the case for C3, C4, I3 (both measures) or C2 (entropy). The interpretation here is not straightforward. It is feasible that the disordered trends can be consequence of non-adaptive processes. An alternative hypothesis is related to the shape of the real distributions for the implementation of any function. We assume that for more and less frequent motifs the analytically deduced probability distributions does not fit equally well the real counterpart. The more abundant the motif, the better the underlying probability distribution is mirrored in its abundance, because the sampling space is covered more readily.
Our analysis of FFLs dynamics was performed considering single, isolated motifs. However, in real systems motifs are embedded in large networks allowing for the combination of motifs. The combination of more abundant motifs, such as C1 and I1, can cover the whole set of possible dynamics by that affecting the abundance of the rest of the motifs.
Evolvability
Assuming that motif plasticity is a relevant trait, our analysis supports the idea that the observed FFL abundance pattern actually correlates with motif evolvability.
Our analysis suggests that neither a direct interpretation of motifs as functional modules [1, 2, 4] nor a purely non-adaptive view of their abundance [22–24] account for the uneven presence in transcription networks. Consistently with previous works [28, 29] duplication-rewiring dynamics alone cannot explain the evolution of FFLs. The potential for evolvability associated to their topological structure might well be the missing ingredient connecting both views.
Conclusions
In this article we have interpreted a simplified, qualitative model of the FFL motif. The thorough analysis within the model framework allows to reconstruct its natural abundance pattern and provides insight in what might have shaped it. The argument leads to the very core of the genotype-phenotype mapping problem, since, due to its simplicity, a perfect mapping between the topology and all possible functions it can implement can be constituted. We claim, however, general applicability. FFL abundances are correlated with their plasticity and evolvability. Evolvability has been defined as a compromise between robustness against single mutations and the capability to modify the functional response upon increasing mutational pressure. The results indicate that a proper portion of intrinsic functional plasticity, which can be understood as a strategic trade-off between specialization and flexibility, is necessary to be abundant. Because only then one is suited to be readily evolvable in changing environments.
Future work should be devoted to analyzing how the coexistence of different motifs embedded in a large network affects their dynamics and abundance compared to the single motif analysis performed in this work.
Methods
Dynamical response of FFL motifs
Here γ_{ i }describes the basal production of protein i, with i = {Y, Z}, subsuming the concentration of all biochemical elements which remain constant in time. The binding equilibrium of the regulators j with the gene G_{ i }are denoted by ${\omega}_{i}^{j}$, with j = {X, Y, Z}. Parameters α^{ x }and β^{ j }define the type of regulatory interactions, i.e. activation or inhibition, for gene G_{ Y }and G_{ Z }, respectively, providing the regulatory rates with respect to the basal transcription. Values < 1 correspond to inhibitory regulation, whereas > 1 accounts for activation (denoted by '-' and '+' in figure 1a respectively). The parameter β^{ xy }accounts for the simultaneous regulation of G_{ Z }. The degradation rate of protein i is denoted as δ_{ i }. Finally, n and m are the degree of multimerization of the regulators.
If we consider the system in phase space, we find that in absence of input the system resides in a stable steady state determined by the crossing of the nullclines, Ẏ = 0 and Ż = 0, respectively. Upon external input, X is activated and hence the shapes of the nullclines change. They provide a new crossing and consequently a new steady state. Due to these changes in the nullclines' geometry the system must evolve from the initial state towards the new stable state. The evolution corresponds to a trajectory crossing phase space that depends on i) the location of the initial state, ii) the location of the final state, iii) the new shapes of the nullclines upon input. The specific dynamics implemented by a given motif is determined by this trajectory, which depends on the set of parameters. However, by analyzing the geometrical features of the nullclines it is possible to determine the so-called Backbone of Requirements for the FFL response (BR), i.e. a set of qualitative relationships between different geometrical features of the nullclines and the location of the initial and the final point that univocally determines the dynamics [26]. Therefore, for a given FFL motif all different sets of parameters satisfying the same BR implement the same function. Similarly, also different BRs may implement the same function.
Conditional probabilities P(ϕ_{ j }| Γ_{ i })
G ^{+} | G ^{-} | P ^{+} T ^{+} | P ^{+} T ^{-} | P ^{-} T ^{-} | P ^{-} T ^{+} | |
---|---|---|---|---|---|---|
C1 | 0.4862 | 0.2111 | 0.2018 | 0.03669 | 0.0642 | 0 |
C2 | 0.0931 | 0.5349 | 0.1861 | 0.09302 | 0.0931 | 0 |
C3 | 0.2336 | 0.5514 | 0.0748 | 0.09346 | 0.0374 | 0.0093 |
C4 | 0.5862 | 0.0689 | 0.1379 | 0.13793 | 0 | 0.0689 |
I1 | 0.3571 | 0.2143 | 0.2857 | 0 | 0.1429 | 0 |
I2 | 0.1111 | 0.4167 | 0.1111 | 0.2222 | 0.0648 | 0.0741 |
I3 | 0.2553 | 0.2766 | 0 | 0.2979 | 0 | 0.1702 |
I4 | 0.4019 | 0.1402 | 0.1869 | 0.1308 | 0.0748 | 0.0654 |
Functional robustness and mutational perturbation
Parametric mutations have different impact on the motif's function, as in nature they can either be neutral or causing qualitative changes. For the system we present here, only those mutations cause functional change, which induce a qualitative alteration in the shape of the nullclines, represented in the Backbone of requirements for the FFL response (BR) [26]. However, the mutation will become visible only, if the resulting BR is actually associated with a different function.
To estimate and compare the degree of mutational robustness for the different FFL motifs, we carried out a numerical study calculating the frequency of functional shifts upon parametric perturbation of equation (6) as shown in figure 5. For a given motif, this can be done introducing random mutations in the parameters that define characteristically the dynamics of FFL motif (figure 5a). We restrict the analysis to that sort of mutations that does not change the topology of the FFL, i.e. mutations that do not change the qualitative type of regulations (activation or inhibition) described by a^{ x }, β^{ x }and β^{ y }.
The suffered mutations are reflected (or not) in a qualitative change of the BR. Here different scenarios are possible, i) mutations that do not affect qualitatively the nullclines' geometry, i.e. there are no changes in the BR (neutral), ii) mutations that are reflected in compatible changes in the BR, but the new BR is associated to the same dynamic than the previous one (neutral), and finally iii) mutations that are reflected in compatible changes in the BR, and the new BR is associated to a different dynamic (qualitatively changing mutations).
In our numerical study we have considered 1000 different sets of parameters for each FFL type (8000 circuits in total). For each FFL, the mutational process is repeated 10.000 times until the probabilities of functional shifts stabilize (figure 5c) and the effects of the accumulation of mutations can be analyzed. This evaluation of the transition probabilities does not depend on specific parameter values but on the conditional relations between them. Since the relations are of the sort a > 6, changes in the function may be achieved by very small or large parameter changes equally likely, depending only on the conditional dependencies between the key-parameters and the corresponding values at time-step t - 1.
Transition probabilities for single mutations C1-C4
G ^{+} | G ^{-} | P ^{+} T ^{+} | P ^{-} T ^{-} | P ^{+} T ^{-} | P ^{-} T ^{+} | |
---|---|---|---|---|---|---|
C1 | ||||||
G ^{+} | 0.251 | 0 | 0.008 | 0 | 0 | 0 |
G ^{-} | 0 | 0.329 | 0 | 0 | 0.003 | 0 |
P ^{+} T ^{+} | 0.015 | 0 | 0.180 | 0.013 | 0 | 0 |
P ^{-} T ^{-} | 0 | 0 | 0.015 | 0.157 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0.0049 | 0 | 0 | 0.031 | 0 |
P ^{-} T ^{+} | 0 | 0 | 0 | 0 | 0 | 0 |
C2 | ||||||
G ^{+} | 0 | 0 | 0.007 | 0 | 0 | 0 |
G ^{-} | 0 | 0.82 | 0 | 0 | 0.007 | 0 |
P ^{+} T ^{+} | 0.008 | 0 | 0.022 | 0.001 | 0 | 0 |
P ^{-} T ^{-} | 0 | 0 | 0.043 | 0.039 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0.007 | 0 | 0 | 0.039 | 0 |
P ^{-} T ^{+} | 0 | 0 | 0 | 0 | 0 | 0 |
C3 | ||||||
G ^{+} | 0.317 | 0 | 0.008 | 0 | 0 | 0 |
G ^{-} | 0 | 0.242 | 0 | 0 | 0.026 | 0 |
P ^{+} T ^{+} | 0.023 | 0 | 0.030 | 0 | 0 | 0 |
P ^{-} T ^{-} | 0 | 0 | 0 | 0 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0.043 | 0 | 0 | 0.174 | 0.046 |
P ^{-} T ^{+} | 0 | 0 | 0 | 0 | 0.046 | 0.151 |
C4 | ||||||
G ^{+} | 0.815 | 0 | 0.019 | 0 | 0 | 0 |
G ^{-} | 0 | 0 | 0 | 0 | 0.019 | 0 |
P ^{+} T ^{+} | 0 | 0 | 0.037 | 0 | 0 | 0 |
P ^{-} T ^{-} | 0 | 0 | 0 | 0 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0.037 | 0 | 0 | 0.037 | 0 |
P ^{-} T ^{+} | 0 | 0 | 0 | 0 | 0 | 0.037 |
Transition probabilities for single mutations I1-I4
G ^{+} | G ^{-} | P ^{+} T ^{+} | P ^{-} T ^{-} | P ^{+} T ^{-} | P ^{-} T ^{+} | |
---|---|---|---|---|---|---|
I1 | ||||||
G ^{+} | 0.241 | 0 | 0.069 | 0 | 0 | 0 |
G ^{-} | 0 | 0.103 | 0 | 0 | 0 | 0 |
P ^{+} T ^{+} | 0 | 0 | 0.345 | 0 | 0 | 0 |
P ^{-} T ^{-} | 0 | 0 | 0 | 0.241 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0 | 0 | 0 | 0 | 0 |
P ^{-} T ^{+} | 0 | 0 | 0 | 0 | 0 | 0 |
I2 | ||||||
G ^{+} | 0.144 | 0 | 0.024 | 0 | 0 | 0 |
G ^{-} | 0 | 0.408 | 0 | 0 | 0.016 | 0 |
P ^{+} T ^{+} | 0 | 0 | 0.160 | 0 | 0 | 0 |
P ^{-} T ^{-} | 0 | 0 | 0 | 0.128 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0.032 | 0 | 0 | 0.080 | 0 |
P ^{-} T ^{+} | 0 | 0 | 0 | 0 | 0 | 0.008 |
I3 | ||||||
G ^{+} | 0.091 | 0 | 0 | 0 | 0 | 0 |
G ^{-} | 0 | 0.212 | 0 | 0 | 0.061 | 0 |
P ^{+} T ^{+} | 0 | 0 | 0 | 0 | 0 | 0 |
P ^{-} T ^{-} | 0 | 0 | 0 | 0 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0.121 | 0 | 0 | 0.394 | 0 |
P ^{-} T ^{+} | 0 | 0 | 0 | 0 | 0 | 0.121 |
I4 | ||||||
G ^{+} | 0.398 | 0 | 0.008 | 0 | 0 | 0 |
G ^{-} | 0 | 0.141 | 0 | 0 | 0.023 | 0 |
P ^{+} T ^{+} | 0 | 0 | 0.063 | 0 | 0 | 0 |
P ^{-} T ^{-} | 0 | 0 | 0 | 0.008 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0.047 | 0 | 0 | 0.156 | 0 |
P ^{-} T ^{+} | 0 | 0 | 0 | 0 | 0 | 0.156 |
Transition probabilities for accumulated mutations C1-C4
G ^{+} | G ^{-} | P ^{+} T ^{+} | P ^{-} T ^{-} | P ^{+} T ^{-} | P ^{-} T ^{+} | |
---|---|---|---|---|---|---|
C1 | ||||||
G ^{+} | 0.212 | 0 | 0.082 | 0.071 | 0 | 0 |
G ^{-} | 0 | 0.235 | 0 | 0 | 0.010 | 0 |
P ^{+} T ^{+} | 0.035 | 0 | 0.113 | 0.052 | 0 | 0 |
P ^{-} T ^{-} | 0.030 | 0 | 0.052 | 0.082 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0.020 | 0 | 0 | 0.007 | 0 |
P ^{-} T ^{+} | 0 | 0 | 0 | 0 | 0 | 0 |
C2 | ||||||
G ^{+} | 0.005 | 0 | 0.008 | 0.006 | 0 | 0 |
G ^{-} | 0 | 0.822 | 0 | 0 | 0.018 | 0 |
P ^{+} T ^{+} | 0.006 | 0 | 0.013 | 0.011 | 0 | 0 |
P ^{-} T ^{-} | 0.008 | 0 | 0.014 | 0.018 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0.056 | 0 | 0 | 0.016 | 0 |
P ^{-} T ^{+} | 0 | 0 | 0 | 0 | 0 | 0 |
C3 | ||||||
G ^{+} | 0.247 | 0 | 0.026 | 0 | 0 | 0 |
G ^{-} | 0 | 0.222 | 0 | 0 | 0.050 | 0.043 |
P ^{+} T ^{+} | 0.016 | 0 | 0.008 | 0 | 0 | 0 |
P ^{-} T ^{-} | 0 | 0 | 0 | 0 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0.100 | 0 | 0 | 0.118 | 0 |
P ^{-} T ^{+} | 0 | 0.086 | 0 | 0 | 0 | 0.086 |
C4 | ||||||
G ^{+} | 0.825 | 0 | 0.069 | 0 | 0 | 0 |
G ^{-} | 0 | 0 | 0 | 0 | 0.005 | 0.005 |
P ^{+} T ^{+} | 0.042 | 0 | 0.011 | 0 | 0 | 0 |
P ^{-} T ^{-} | 0 | 0 | 0 | 0 | 0 | 0 |
P ^{ + } T ^{-} | 0 | 0.011 | 0 | 0 | 0.011 | 0 |
P ^{ - } T ^{+} | 0 | 0.011 | 0 | 0 | 0 | 0.011 |
Transition probabilities for accumulated mutations I1-I4
G ^{+} | G ^{-} | P ^{+} T ^{+} | P ^{-} T ^{-} | P ^{+} T ^{-} | P ^{-} T ^{+} | |
---|---|---|---|---|---|---|
I1 | ||||||
G ^{+} | 0.171 | 0 | 0.114 | 0.086 | 0 | 0 |
G ^{-} | 0 | 0.024 | 0 | 0 | 0 | 0 |
P ^{+} T ^{+} | 0.033 | 0 | 0.229 | 0.098 | 0 | 0 |
P ^{-} T ^{-} | 0.024 | 0 | 0.098 | 0.122 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0 | 0 | 0 | 0 | 0 |
P ^{-} T ^{+} | 0 | 0 | 0 | 0 | 0 | 0 |
I2 | ||||||
G ^{+} | 0.088 | 0 | 0.048 | 0.040 | 0 | 0 |
G ^{-} | 0 | 0.398 | 0 | 0.007 | 0.029 | 0.007 |
P ^{+} T ^{+} | 0.016 | 0 | 0.088 | 0.040 | 0 | 0 |
P ^{-} T ^{-} | 0.013 | 0.015 | 0.040 | 0.061 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0.058 | 0 | 0 | 0.037 | 0 |
P ^{-} T ^{+} | 0 | 0.015 | 0 | 0 | 0 | 0.001 |
I3 | ||||||
G ^{+} | 0.024 | 0 | 0 | 0 | 0 | 0 |
G ^{-} | 0 | 0.165 | 0 | 0 | 0.098 | 0.039 |
P ^{+} T ^{+} | 0 | 0 | 0 | 0 | 0 | 0 |
P ^{-} T ^{-} | 0 | 0 | 0 | 0 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0.196 | 0 | 0 | 0.353 | 0 |
P ^{-} T ^{+} | 0 | 0.078 | 0 | 0 | 0 | 0.047 |
I4 | ||||||
G ^{+} | 0.401 | 0 | 0.033 | 0.017 | 0 | 0 |
G ^{-} | 0 | 0.088 | 0 | 0 | 0.032 | 0.032 |
P ^{+} T ^{+} | 0.037 | 0 | 0.035 | 0.005 | 0 | 0 |
P ^{-} T ^{-} | 0.009 | 0 | 0.003 | 0.001 | 0 | 0 |
P ^{+} T ^{-} | 0 | 0.064 | 0 | 0 | 0.088 | 0 |
P ^{-} T ^{+} | 0 | 0.064 | 0 | 0 | 0 | 0.088 |
where ${\sum}_{i}{\omega}_{k}^{m}\left({\varphi}_{i}|{\varphi}_{i}\right)$ represents the robustness against accumulated and ${\sum}_{i}{\Omega}_{k}^{1}\left({\varphi}_{i}|{\varphi}_{i}\right)$ represents the robustness against single mutations. Since ${\sum}_{i}{\Omega}_{k}^{1}\left({\varphi}_{i}|{\varphi}_{i}\right)>{\sum}_{i}{\Omega}_{k}^{m}\left({\varphi}_{i}|{\varphi}_{i}\right)$, we have $0\le \mathcal{E}\left({\Gamma}_{k}\right)\le 1$.
Entropy, kurtosis and the likelihood of appearance of FFL motifs
where < P > is the mean value of the probability distribution and σ is the standard deviation.
The calculated kurtosis values are K(Γ_{ s }) = 6 and K(Γ_{ f }) = -3.33, knowing that n = 6 (number of different possible functions).
Again, we will first consider the extreme cases Γ_{ s }and Γ_{ f }to determine the range of all possible values. We find for the most flexible case P(ϕ_{ j }| Γ_{ f }) = 1/6 an entropy of H(Γ_{ f }) = log_{2}(6). For the most specialized case Γ_{ s }(P(ϕ_{1} | Γ_{ s }) = 1 and 0 otherwise) the associated entropy is H(Γ_{ s }) = 0. Any FFL motif will have entropy values residing within this range. When correlating the FFLs' entropy and their abundance in figure 4b we find that the most abundant motifs show intermediate values. This trend coincides with what has been found when applying kurtosis, where too, C1 and I1 show intermediate kurtosis values. They do neither exhibit high flexibility nor high specialization, indicating that a trade-off between both features is associated to the most abundant motifs. It can be understood in terms of adaptability or in other words the motifs' plasticity to explore the landscape of possible dynamics under mutational pressure.
Declarations
Acknowledgements
We thank the members of the Complex Systems Lab for fruitful discussion. This work was supported by the EU grant CELLCOMPUT, the EU 6th Framework project SYNLET (NEST 043312), the James McDonnell Foundation, the Marcelino Botín Foundation, the University of Vienna and by the Santa Fe Institute.
Authors’ Affiliations
References
- Kirschner M, Gerhart J: Evolvability. Proc Natl Acad Sci USA. 1998, 95: 8420-7. 10.1073/pnas.95.15.8420.View ArticleGoogle Scholar
- Wagner A: Robustness and evolvability: a paradox resolved. Proc R Soc London B. 2008, 275: 91-100. 10.1098/rspb.2007.1137.View ArticleGoogle Scholar
- Ancel LW, Fontana W: Plasticity, evolvability, and modularity in RNA. J Exp Zool. 2000, 288: 242-83. 10.1002/1097-010X(20001015)288:3<242::AID-JEZ5>3.0.CO;2-O.View ArticleGoogle Scholar
- Stadler B, Stadler P, Wagner G, Fontana W: The topology of the possible: formal spaces underlying patterns of evolutionary change. J theor Biol. 2001, 213: 241-274. 10.1006/jtbi.2001.2423.View ArticleGoogle Scholar
- Alon U: Network motifs: theory and experimental approaches. Nature Reviews Genetics. 2007, 8: 450-61. 10.1038/nrg2102.View ArticleGoogle Scholar
- Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U: Network motifs: simple building blocks of complex networks. Science. 2002, 298: 824-7. 10.1126/science.298.5594.824.View ArticleGoogle Scholar
- Shen-Orr SS, Milo R, Mangan S, Alon U: Network motifs in the transcriptional regulation network of Escherichia coli. Nat Genet. 2002, 31: 64-8. 10.1038/ng881.View ArticleGoogle Scholar
- Mangan S, Zaslaver A, Alon U: The Coherent Feedforward Loop Serves as a Sign-sensitive Delay Element in Transcription Networks. J Mol Biol. 2003, 334: 197-204. 10.1016/j.jmb.2003.09.049.View ArticleGoogle Scholar
- Wall M, Dunlop MJ, Hlavacek W: Multiple Functions of a Feed-Forward-Loop Gene Circuit. JMB. 2005, 349: 501-14. 10.1016/j.jmb.2005.04.022.View ArticleGoogle Scholar
- Ozbudak EM, Thattai M, Lim HN, Shraiman BI, Oudenaarden AV: Multistability in the lactose utilization network of Escherichia coli. Nature. 2004, 427: 737-40. 10.1038/nature02298.View ArticleGoogle Scholar
- Alon U, Surette MG, Barkai N, Leibler S: Robustness in bacterial chemotaxis. Nature. 1999, 397: 168-71. 10.1038/16483.View ArticleGoogle Scholar
- Barkai N, Leibler S: Robustness in simple biochemical networks. Nature. 1997, 387: 855-7. 10.1038/43072.View ArticleGoogle Scholar
- Ishihara S, Fujimoto K, Shibata T: Cross talking of network motifs in gene regulation that generates temporal pulses and spatial stripes. Genes Cells. 2005, 10: 1025-1038. 10.1111/j.1365-2443.2005.00897.x.View ArticleGoogle Scholar
- Cotterell J, Sharpe J: An atlas of gene regulatory networks reveals multiple three-gene mechanisms for interpreting morphogen gradients. Mol Sys Biol. 2010, 6: 425-438.Google Scholar
- Mangan S, Alon U: Structure and function of the feed-forward loop network motif. Proc Natl Acad Sci USA. 2003, 100: 11980-5. 10.1073/pnas.2133841100.View ArticleGoogle Scholar
- Kalir S, Mangan S: A coherent feed-forward loop with a SUM input function prolongs flagella expression in Escherichia coli. Mol Syst Biol. 2005, 1: 2005.0006Google Scholar
- Draghi J, Wagner GP: Evolution of evolvability in a developmental model. Evolution. 2008, 62: 301-15. 10.1111/j.1558-5646.2007.00303.x.View ArticleGoogle Scholar
- Prill RJ, Iglesias PA, Levchenko A: Dynamic properties of network motifs contribute to biological network organization. PLoS Biol. 2005, 3: e343-10.1371/journal.pbio.0030343.View ArticleGoogle Scholar
- Ghosh B, Karmakar R, Bose I: Noise characteristics of feed forward loops. Phys Biol. 2005, 2: 36-45. 10.1088/1478-3967/2/1/005.View ArticleGoogle Scholar
- Cordero OX, Hogeweg P: Feed-forward loop circuits as a side effect of genome evolution. Mol Biol Evol. 2006, 23: 1931-6. 10.1093/molbev/msl060.View ArticleGoogle Scholar
- Ingram PJ, Stumpf MP, Stark J: Network motifs: structure does not determine function. BMC Genomics. 2006, 7: 108-10.1186/1471-2164-7-108.View ArticleGoogle Scholar
- Lynch M: The evolution of genetic networks by nonadaptive processes. Nat Rev Genet. 2007, 8: 803-813.View ArticleGoogle Scholar
- Solé RV, Valverde S: Are network motifs the spandrels of cellular complexity?. Trends Ecol Evol. 2006, 21: 419-22. 10.1016/j.tree.2006.05.013.View ArticleGoogle Scholar
- Solé RV, Valverde S: Spontaneous emergence of modularity in cellular networks. J R Soc Interface. 2008, 5: 129-33. 10.1098/rsif.2007.1108.View ArticleGoogle Scholar
- Mazurie A, Bottani S, Vergassola M: An evolutionary and functional assessment of regulatory network motifs. Genome Biol. 2005, 6: R35-10.1186/gb-2005-6-4-r35.View ArticleGoogle Scholar
- Macía J, Widder S, Solé RV: Specialized or flexible feed-forward loop motifs: a question of topology. BMC Syst Biol. 2009, 31: 3-84.Google Scholar
- Mangan S, Itzkovitz S, Zaslaver A, Alon U: The Incoherent Feed-forward Loop Accelerates the Response-time of the gal System of Escherichia coli. J Mol Biol. 2006, 356: 1073-81. 10.1016/j.jmb.2005.12.003.View ArticleGoogle Scholar
- Conant GC, Wagner A: Convergent evolution of gene circuits. Nat Genet. 2003, 34: 264-266. 10.1038/ng1181.View ArticleGoogle Scholar
- Teichmann SA, Babu MM: Gene regulatory network growth by duplication. Nat Genet. 2004, 36: 492-496. 10.1038/ng1340.View ArticleGoogle Scholar
- Balanda KP, MacGillivray HL: Kurtosis: a critical review. Am Statistn. 1988, 42: 111-9. 10.2307/2684482.Google Scholar
- Spivak M: Calculus. 1994, Cambridge University Press, 201-11.Google Scholar
- Shannon CE: A Mathematical Theory of Communication. Bell System Technical Journal. 1948, 27: 379-423.View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.