September 26, 2012

Cascades in Common: biological network function in evolution

What is a network cascade? And what is their relevance to biological evolution? The answer, while not definitive in this post, is that they contribute to both adaptive variation and variation between species. Cascades have been defined in a range of complex networks, from economic systems and social hierarchies to cellular processes and the internet. In [1], cascades are defined as the downstream effects of some disruption or change in the network (termed higher-order interrelationships). Of particular interest are small, localized disruptions that result in widespread state changes (such as innovations, death, and revolutions) [2]. While these type of responses are not typical network behavior (as we will see), cascade dynamics can be altered by changing the connectivity of a network [3]. In [4], nodes and edges were added to maximize the spread of cascade activity in a set of network topologies. Thus, cascades can also result from the "bursty" nature of information and activity flows in a network, as the activity of network components is often stochastic and rarely coordinated.

Figure 1. A: Example of a link (edge) in a genetic regulatory network (arrow c is linked to gene g via activation of binding site n). B: a kinetic model that describes the relationship between the occupancy of n and the input c given four possible outcomes. C: effect of gene network on a phenotype (A-P axis of Drosophila during development). COURTESY: Figures 1, 2, and 10 in [5].

The complex and subtle effects and dynamics of cascades cannot be understood outside the context of biological networks. In this case, we will focus on networks representing gene regulation and more general physiological mechanisms. To better understand the outcomes of gene expression in organisms, researchers have used an approach called genetic regulatory networks (GRNs). GRNs are hierarchical, directed networks that are modular [6], exhibit differential effects [6], and feature a number of important topological motifs [7]. Hierarchical gene regulatory networks are rooted (top level) by master regulatory genes, which code for transcription factors [8] that regulate many downstream targets. The effects of connectivity can range from immediate (first-order) connections to more diffuse (second- and third-order) connections [9]. An example focusing on second-order connectivity can be seen in Figure 2. While the connectivity between nodes is an important determiner of function, the flows between nodes (or edges) are also an important feature of gene regulatory networks. Flows (or slow cascades) are perhaps a better indicator of functional potential, as different parts of the network can be activated or inactivated in different contexts. These flows contribute to activation of the network from top to bottom, which can be characterized as a cascade [9]. 

Figure 2. Two examples of second-order interconnectivity in a hierarchical, directed graph. COURTESY: Figure 3 in [1].

How can we understand the role of cascades in evolution, and how are they affected by evolution? In Erwin and Davidson [6], the effects of changes in a given GRN architecture depend on where in the network these changes occur. In a recent review by Cohen [10], the concept of physiological regulatory networks (PRNs) is introduced. In PRNs (see Figure 3), nodes can represent a wide range of physiological phenomena (e.g. immune molecules, metabolites), each class of which constitutes a subnetwork. Notably, homeostasis [11] governs the state of edges between these nodes in a way that regulates the global network state. There is an interesting interaction between homeostasis and the propagation of cascades, especially as it relates to contributions of heterogeneous subnetworks towards the global physiological state. Using this formulation of a network, alternative physiological structures may form, which are different network topologies that have equivocal functional outcomes and the same overall fitness [10]. This provides a degree of robustness that protects the organisms from potentially disastrous changes to the network structure. This is also true for many developmental processes, as networks are conserved across phylogeny [8]. However, changes to developmental networks may also enable evolutionary changes. In the case of one such process (the oocyte-to-embryo transition), certain genes can lose or gain function which can in turn enable reproductive isolation [11]. Thus, the network properties that enable cascades to occur could also enable evolutionary change.

Figure 3. An example of a PRN topology and its outputs. COURTESY: Figure 1 in [10].

Using a strictly hierarchical gene regulatory network, I would like to conduct a thought experiment to illustrate the changes required and observed across phylogeny, where we begin with a densely connected graph and then delete connections one by one in the fashion of a sensitivity analysis. The deletions could represent mutational perturbations or some other unspecified disruption. In this example, suppose that we have a fully-connected random topology with activation spreading from the top nodes to the bottom (see Figure 4, right). Because it is fully connected, this topology exhibits maximal robustness, and as such cannot be highly disrupted unless many edges are removed. Now let us consider a network topology that is sparsely connected at the top with activation spreading in the same manner as before (Figure 4, left). In this case, large parts of the network can be inactivated with the removal of many fewer connections. Because it has fewer redundant connections, there are fewer opportunities to re-route the influence from higher levels.

Figure 4. Three-level strictly hierarchical gene regulatory network from thought experiment.

It is of note that in both cases, nodes are treated as static entities. Interestingly, a study by LeClerc [13] suggests that in evolutionary simulations, sparser network topologies tend to evolve from an initial condition of dense connectivity. This is because the resulting sparse topologies retain the dynamic robustness of the original topology, a result also observed in a wide range of species (e.g. yeast, Arabidopsis, and Drosophila among others). In terms of cascades, Duncan Watts suggests that in general, the propensity for large-scale cascades to be triggered by perturbations of varying magnitude is a nonlinear response [14]. Large-scale cascades, or avalanche-like displacements, occur rather infrequently and define major functional events or changes in the biological system [15]. Furthermore, cascades can either be progressive (propagate at a regular rate) or non-progressive (propagate irregularly by chance) [3], which can either provide opportunities for an adaptive response, or act as a set of constraints on network topology evolution. As in the case of LeClerc's evolutionary simulations, the propensity of cascades in a network also occurs with regard to connectivity. In sparsely connected networks, the perturbation of highly-connected nodes is most likely to trigger cascades. In densely- connected and heterogeneous networks, the propagation of cascades are limited to the local stability of nodes. Cascades triggered in this manner are very unstable and harder to predict, and point to potential roles for transcriptional stability and noise in cascade dynamics.

What does this tell us about evolutionary systems? Is this a secondary reason why developmental networks tend to be conserved (the first being to "lock-in" essential function)? And what happens when a large-scale cascade occurs in a gene regulatory or physiological network? To get at these questions, we must understand what kinds of changes are enabled when key features of a network are changed. In Bhardwaj, Yan, and Gerstein [16], the hierarchical layout of a GRN is compared to social hierarchies. In particular, GRNs can be either autocratic (generally sparsely connected from top to bottom) or democratic (more densely connected and less dependent on higher levels). Examples of this range from the expansion of existing hierarchical layers of regulation in E.coli genomes (e.g. democratic) to differences between top-level and mid-level transcription factors in yeast genomes (e.g. more autocratic) [17]. The hubs and other structural features of regulatory hierarchies also provide an opportunity for coordinating global changes in gene expression based on relatively few mutational changes and enhancing the effects of beneficial mutations [17]. 

Using models of this type, the potential for cascades (as opposed to normal GRN function) can be modeled using computational simulations of sea urchin embryonic development [18]. This exercise demonstrates that transitions between regulatory "states" can occur sharply, which is contingent upon connectivity (e.g. more likely in autocratic networks). Secondly, the model of Boulouri and Davidson [18] shows that each hierarchical level is activated in succession, with a lag of a few hours for each level activated. Perturbations of and significant delays in this lag time (through a variety of mechanisms) could lead to large-scale evolutionary changes via developmental timing and coherence.

Perhaps cascades and changes in network connectivity due to perturbation (environmental or mutational) indeed play a role in setting up evolutionary changes. At the very least, the interaction between standing variation, adaptive changes, and genetic/physiological regulation can be better understood by considering the role of cascade dynamics. This is a largely unexplored area, but future work in the areas of dynamical evolutionary simulations, network representations, and multilevel/multiscalar models may provide us with some useful advances.

[1] Acemoglu, D., Asuman Ozdaglar, A., and Tahbaz–Salehi, A.   Cascades in Networks and Aggregate Volatility. NBER Working Paper 16516 (2010).

[2] Culotta, A.   Maximizing Cascades in Social Networks: an overview. Technical Report, Computer Science Department, University of Massachusetts, Amherst (2003).

[3] Sheldon, D., Dilkina, B., Elmachtoub, A., Finseth, R., Sabharwal, A., Conrad, J., Gomes, C., Shmoys, D., Allen, W., Amundsen, O., and Vaughan, B.   Maximizing the Spread of Cascades Using Network Design. arXiv, 1203.3514 (2012).

[4] Feldmann,  A., Gilbert, A.C. and  Willingert, W.   Data networks as cascades: investigating the multifractal nature of Internet WAN traffic. Proceedings of SIGCOMM (1998).

For information on potential natural defense mechanisms against large-scale cascades that can severely disrupt a network's function, please see: Motter, A.E.   Cascade control and defense in complex networks. Physical Review Letters, 93, 098701 (2004).

[5] Tkacik, G. and Walczak, A.M.   Information transmission in genetic regulatory networks: a review. Journal of Physics: Condensed Matter, 23, 153102 (2011).

[6]  Erwin, D.H. and Davidson, E.H.   The evolution of hierarchical gene regulatory networks. Nature Reviews Genetics, 10, 141-148 (2009).

[7] Ingram, P.J., Stumpf, M.P.H., and Stark, J.   Network motifs: structure does not determine function. BMC Genomics, 7, 108 (2006). 

This paper demonstrates that the bi-fan motif (a type of one-to-many hierarchical relationship) has a broad range of potential functional responses.

[8] Gehring, W.J.   Master Control Genes in Development and Evolution: the Homeobox story. Yale University Press, New Haven (1998). 

For a basic definition, please see: Myers, P.Z.   Master Control Genes and Pax-6. Pharyngula, (2007)

Examples of master regulatory genes include Oct4 (pluripotency), Pax6 (eye development), and p53 (cancer).

[9] Watts, D.J.   A simple model of global cascades on random networks. PNAS, 99(9), 5766 –5771 (2002).

[10] Cohen, A.A., Martin, L.B., Wingfield, J.C., McWilliams, S.R., and Dunne, J.A.   Physiological regulatory networks: ecological roles and evolutionary constraints. Trends in Ecology and Evolution, 27(8), 428-435 (2012).

[11] Cannon, W.   Wisdom of the Body. W.W. Norton, New York (1930). 

For a basic definition, please see this Scientific American article (

[12] Evsikov, A.V., Graber, J.H., Brockman, J.M., Hampl, A., Holbrook, A.E., Singh, P., Eppig, J.J., Solter, D., and Knowles, B.B.   Cracking the egg: molecular dynamics and evolutionary aspects of the transition from the fully grown oocyte to embryo. Genes and Development, 20, 2713–2727 (2006).

[13] LeClerc, R.D.   Survival of the sparsest: robust gene networks are parsimonious. Molecular Systems Biology, 4, 213 (2008).

[14] Watts, D.J.   A simple model of global cascades on random networks. PNAS, 99(9), 5766 –5771 (2002).

Large-scale cascades (e.g. avalanches) can be caused by relatively small perturbations. In technical terms, the size distribution of cascades relative to the degree of perturbation is variable according to a power law. Also see related concept of scale-free networks.

[15] Pajevic, S. and Plenz, D.   Efficient Network Reconstruction from Dynamical Cascades Identifies Small-World Topology of Neuronal Avalanches. PLoS Computational Biology, 5(1), e1000271 (2009).

[16] Bhardwaj, N., Yan, K-K., and Gerstein, M.B.   Analysis of diverse regulatory networks in a hierarchical  context shows consistent tendencies for collaboration in the middle levels. PNAS, 107(15), 6841–6846 (2010).

[17] Crombach, A. and Hogeweg, P.   Evolution of Evolvability in Gene Regulatory Networks. PLoS Computational Biology, 4(7), e1000112 (2008).

[18] Bolouri, H. and Davidson, E.H.   Transcriptional regulatory cascades in development: initial rates, not steady state, determine network kinetics. PNAS, 100(16), 9371–9376 (2003).

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