Evolvability of feed-forward loop architecture biases its abundance in transcription networks

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.

Figure 1

Structure and frequency of FFL motifs. In (a) we show the schematic representation of the FFL's genetic regulatory interactions ('+' represents activatory regulation and '-' represents inhibitory regulation). The external input I activates the signal protein X. Active X modulates expression of gene G_Z directly and indirectly via regulation of Y expression, which in turn also regulates G_Z. The dynamics of these regulatory interactions is described by a set of equations dy/dt = F(y, z), dz/dt = G(y, z) incorporating the nonlinearities associated to gene-gene interactions. In (b) we plot the general topology of FFL motifs and the six different functions ϕ(t) represented by qualitative time-courses [Z(t)]. 'G' indicates grader dynamics, 'P' pulser dynamics. We specifically take into account the initial slope of the time-course ('+' or '-') and the concentration of the final target Z with respect to the non-induced protein concentration ('T+' and 'T-'). In (c) we display the relative abundance P_obs(Γ_i) of these motifs in the transcription networks of yeast and E. coli (data from [27]).

Figure 2

FFL function and their probabilities. (a) The landscape of FFL motifs is displayed as a bipartite graph linking patterns (upper row) and processes (lower). The weight of the links indicates the relative probability P(ϕ_j | Γ_i) that a given motif Γ_i implements a given function ϕ_j. In (b) the matrix of motif-function probabilities is displayed using a color scale. The plots highlight that some motifs look more specialized, whereas others display rather evenly distributed functional responses.

Figure 3

Predicted probability and FFL abundance. In (a) we compare the natural abundance and its predicted counterpart ρ(Γ_i). S. cerevisiae (black box) is compared to E. coli (white box) and the predicted probabilities (black triangle). In (b) we present the correlation between ρ(Γ_i) and the natural abundances. The Pearson coefficient for the linear fit is r = 0.91 and r = 0.94 for E. coli and S. cerevisiae, respectively.

Figure 4

FFL plasticity and abundance. Here we compare the measures kurtosis, entropy and ψ(Γ_i) and their correlation with the abundance of NW motifs (S. cerevisiae shown in black and E.coli in white). In (a) the kurtosis of the motif's probability distribution for different functions versus the motifs abundance is plotted. In (b) we show entropy versus abundance. The most abundant motifs have intermediate values for both measures which can be interpreted as high plasticity in both cases. In (c) we present the correlation between ψ(Γ_i) and abundance. ψ(γ_i) correlates negatively and thus again, plasticity correlates positively with motif abundance.

Figure 5

Computation of the transition frequencies. Sketch of the procedure. In (a) the update rule is shown. For a given string of conditional relations (BR) the associated dynamical pattern is calculated at time-step t - 1. Next the entries of BR are mutated at time-step t and the (new) dynamical pattern is evaluated. Then the transition of pattern_t-1 to pattern_t is binned. This protocol is executed until no more changes in the bins occur as shown in plot (c). In (b) the graph Ω_m(Γ_i) associated to the transitions between the possible types of dynamics is represented for C1. The thickness of the arrows correspond to transition probabilities obtained from procedure (a). From these graphs (see Methods) we can calculate the motif evolvability ε, which is found to positively correlate with ρ(Γ_i).

Figure 6

FFL evolvability. In plot (a) we show the correlation between the motif's evolvability and its abundance. Evolvability ε, which is found to positively correlate with ρ(Γ_i), is highest for the most abundant motifs C1, I1. In black we show data point of S. cerevisiae, in white E. coli. In (b) we show the correlation between the FFL's evolvability and its ψ(Γ_i). We calculate a Pearson coefficient of r = -0.92 for the linear fit. The lower ψ(Γ_i), the higher the motif's plasticity and the higher its evolvability. The data points are developed from the motifs topology and thus are species independent (blue).