2.1 Toy model
To illustrate the predictable behaviour of monotone systems, we compared two versions of a feedforward loop, one sign consistent and the other inconsistent. Using a system of ODEs, we analysed the effects of varying rates of interactions on the levels of output node as a function of time. For sign-consistent feedforward loops, irrespective of the rates, the output behaviour is always predictable. The output node D always increased monotonically (). On the other hand, when the signs of the links in the loop are inconsistent, the behaviour is not predictable. Different rates can make the output node D increase or decrease monotonically depending on the actual parameters ().
Simulation of toy models of positive and negative feedforward loop
2.2 Counting feedback loops
All three regulatory networks that were analysed have similar nodes to links ratio, positive to negative links ratio and display ‘small-world’ properties (high clustering coefficients and similar characteristic path lengths compared with random networks) (). All three networks have a similar connectivity distribution that best fit a power-law function (). This indicates that the networks are enriched in highly connected nodes (hubs) in comparison to random networks, and as such the regulatory networks contain genes and proteins that are directly regulating or are regulated by many other nodes, or both. Counting the number of positive against negative feedback and feedforward loops in the real regulatory networks against shuffled networks, we find that there are significantly more positive loops as compared with loops found in the three types of shuffled networks (). Using approximate binomial distribution analysis, we explain that these results are not due to the fact that the networks contain more positive than negative links.
Characteristics of the real regulatory networks
Comparison of loops in real and corresponding randomised networks
If there are P positive and N negative links, and P and N are sufficiently large, the probability of picking, using a Bernoulli process, a negative link is p(−) = N/(N + P) and a positive link is p(+) = 1 − p(−).
We define ploop(k) as the probability that a feedback or feedforward loop is positive, where k is the number of links and nodes making up the loop. A positive loop is defined as a loop with either all positive links or an even number of negative links. Thus, we have the following linear first-order recursion
This recursion has the solution
Thus, for 0 < p(+) < 1, ploop(k) converges to 0.5, and for p(+) = 1 ploop(k) = 1 (all links are positive) and ploop(k) alternates between 0 and 1 if p(+) = 0 (the network is made of only negative links).
For example, for k = 5 and the E. coli transcriptional network, where we have 321 positive links, 172 negative links, and 26 neutral links (we count neutral links as positive) we have p(+) = 347/(347 + 172) = ~0.67.
Therefore using this simplified Bernoulli argument, the probability of obtaining a positive loop is ploop
(5) = [1 + (0.34)5
]/2 = 0.502 (for k
= 4, p
= 0.507, and for k
= 3, p
= 0.52). This analysis is similar to what was suggested for the different possible configurations of three-node feedforward loops in a prior study [19
The real networks were compared with three types of shuffled networks used as statistical controls: the first type of shuffled networks maintains the exact connectivity as the natural topology but differ in the distribution of signs/effects associated with the links (sign-swapped). The other two types of shuffled networks are Erdos–Renyi random networks [20
] which are completely randomised and Maslov–Milo method of shuffling which preserves some of the original topology [4
]. The Maslov–Milo shuffling method maintains the connectivity distribution of nodes but destroys the local structure of the networks by repeatedly swapping the connectivity of pairs of interactions making source nodes randomly linked to target nodes. The Maslov–Milo shuffling method does not preserve the sign distribution by blindly swapping positive and negative links.
Interestingly, the difference between the real and sign-swapped shuffled networks for the yeast transcription regulatory network is less significant than differences observed from similar comparisons for the mammalian signalling and E. coli
transcriptional networks (). This difference can be explained by a single link that is found in many feedforward loops in the yeast network. The positive link between DAL80 and GLN3, when removed, caused the abolishment of 22 out of a total of 42 negative loops. Both genes encode GATA family transcription factors, where DAL80 is a repressor that functions as an outgoing hub that is regulated positively by GLN3 [22
]. The GATA family of genes makes an extensive regulatory circuit which contains many members of the family regulating one another [23
]. The mathematical derivation described above assumes statistical independence of link contribution to loops and as such would result in almost even number of positive against negative loops in shuffled networks. This is observed for the signed-shuffled E. coli
gene regulatory and mammalian cell signalling networks. In contrast, in the yeast gene regulatory network, a single link participates in 22 negative loops. Removing this link abolishes all these loops. We term this phenomenon of shared links (and/or nodes) to form multiple loops ‘nesting’. Nesting can drastically affect the distribution of positive to negative loops ratio. In the natural topologies, as well as in the sign-swapped topologies, there are several links that contribute to the formation of many negative loops in each network ().
Links reused in feedback and feedforward loops
2.3 Removing negative loops
We developed an algorithm that removes links that contribute to the formation of negative feedback and feedforward loops. The algorithm is demonstrated schematically on a toy network (). We sequentially remove links to gradually eliminate all small size negative loops (3–4 or 3–5 nodes per loop) from any network. We applied this algorithm to the original networks and compared the results with the results of applying the algorithm to random-swap shuffled networks. We found that we could remove ~30% less links from the E. coli transcription network, ~50% less links from the yeast transcription network or ~65% less links from the mammalian signalling network, compared with the number of links that need to be removed from the corresponding shuffled networks in order to eliminate all negative loops of size 3–5 for the yeast and E. coli networks and 3–4 for the signalling network (). These results indicate that it is easier to convert real network topologies to monotone ‘sign-consistent’ topologies as compared with the ‘effort’ needed to convert shuffled networks into monotone topology.
Toy network to illustrate negative loops removal algorithm
Gradual removal of links that contribute to negative loops
2.4 Clumping of negative links by hubs
The relative abundance of positive loops in the real networks and the relative ease in removing the negative loops could be due to few highly connected nodes that have predominantly negative links. Since hubs contribute to the formation of many feedback and feedforward loops and if a hub has all or mostly negative links, the probability that such hub would contribute two negative links to a loop is high. Since each node in a feedforward or feedback loop contributes two links to the loop, the two negative links in a loop that are connected to a ‘negative’ hub would cancel each other, because, by definition two adjacent negative links are considered positive. This feature of the network would make hubs with many negative links (negative hubs) sequester the negative links because these negative hubs would contribute pairs of negative links to loops. Pairs of negative links in loops that involve negative hubs would make these loops positive because of the double negative. This would leave only few sparsely connected nodes with negative links that can be present in negative loops.
Consider a graph G. We select one node as our ‘hub’ and divide the graph into two graphs: G1 is the hub with its edges (a star-like graph) and G2 is the rest of the graph (the hub’s neighbouring nodes appear in both graphs). There is a fraction P2 of positive links in G2 and a fraction n2 = 1 − p2 of negative links. We assume that p2 > 1/2, that is, there are more positive links than negative links in G2. This assumption holds for the networks we analysed, and if we chose a hub with relative majority of negative edges than the fraction of positive edges in G2, this makes our assumption even stronger than that of G.
We denote by σk
the probability of a simple path of length k
to be positive (i.e. to include even number of negative links). σ1
Assuming p2 > 1/2, we obtain σ1 > 1/2 and σ2 > 1/2. We show now that σk ≥ 1/2 for any k. We have shown this for k = 1 and 2, and now we want to show that if it is true for k then it must be true for k + 1 as well.
A path of length k + 1 is a path of length k with one additional link. It can be positive in one of the two ways: either the original path (of length k) is positive, and the extra edge is also positive or both are negative. Thus, σk+1 = p2 · σk + (1 − p2) · (1 − σk). Now by simple algebra
According to our assumptions, σk ≥ 1/2 which means that 2σk − 1 ≥ 0. In addition, 1 − p2 < 1/2 and as a result: −σk+1 < (2σk − 1)/2 − σk = −1/2 or σk+1 ≥ 1/2.
Now we want to close this path into a loop, using two edges from G1. We denote the fraction of positive edges in G1 by p. The fraction of negative edges is n = 1 − p.
The probability ploop of the loop to be positive is
We showed already that σk ≥ 1/2 for any k and thus 2σ − 1 ≥ 0. Thus the second term is negative. The function f (p) = p(1 − p) obtains its maxima at p = 1/2 and is equal to zero at p = 0 and 1. Thus, ploop obtains its minima when the hub is balanced (p = 1/2) and is increasing as p approaches 0 or 1.
To determine whether real networks are enriched in negative hubs, we first plotted the in-links against out-links difference on the x-axis and positive-links against negative-links difference on the y-axis for all nodes (). The plots show the existence of hubs with abundance of negative links in all three real networks compared with a representative random-swap shuffled network. In particular, the yeast and the E. coli transcriptional networks have many more outgoing negative (and also positive) hubs. All three intracellular regulatory networks show preferential enrichment for hubs with either only positive or only negative links. Because negative links are concentrated around few hubs, and are not evenly spread around like in the shuffled networks, the probability of forming negative loops is reduced. Hence, the existence of hubs enriched in negative links in the topology of real networks leads to the preference for positive feedback and feedforward loops.
Visualisation of positive and negative and in-out hubs