Identifying robust hysteresis in networks

<div><p>We present a new modeling and computational tool that computes rigorous summaries of network dynamics over large sets of parameter values. These summaries, organized in a database, can be searched for observed dynamics, e.g., bistability and hysteresis, to discover parameter regimes over which they are supported. We illustrate our approach on several networks underlying the restriction point of the cell cycle in humans and yeast. We rank networks by how robustly they support hysteresis, which is the observed phenotype. We find that the best 6-node human network and the yeast network share similar topology and robustness of hysteresis, in spite of having no homology between the corresponding nodes of the network. Our approach provides a new tool linking network structure and dynamics.</p></div

Computational time.

<p>CPU time to compute all four resettable bistability and hysteresis queries for various networks.</p

3 node E2F-Rb networks.

<p>(a) 3 node network with potential edges from [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006121#pcbi.1006121.ref004" target="_blank">4</a>] with signal <i>S</i> acting on <i>MD</i>. (b) Histogram showing 24 regulatory networks expressing full path hysteresis between QS and PS. Five regulatory networks show full path hysteresis in 100% of <i>EPG</i>(¬<i>MD</i>). (c) Histogram showing 27 regulatory networks expressing partial path hysteresis between QS and PS. (d) Histogram showing 25 regulatory networks expressing full path resettable (QS,PS) bistability to QS. (e) Histogram showing 35 regulatory networks expressing full path resettable (QS,PS) bistability to QS. (f)-(j) Five networks that exhibit full path hysteresis along all appropriate paths in <i>EPG</i>(¬<i>MD</i>) These networks are also the top five networks in partial path hysteresis in descending order (f),(g)-(i), (j). Network (f) is also highest in prevalence of partial path resettable bistability, but does not show any full path resettable bistability. Full results for all networks can be find in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006121#pcbi.1006121.s001" target="_blank">S1 Table</a>, where the five networks (f)-(j) have numbers 24,39,46,26 and 22.</p

Hysteresis and resettable bistability.

<p>Solid (dashed) lines indicate stable (unstable) equilibria for fixed values of input, Off = Low Equilibrium, B = Bistability, and On = High Equilibrium. <i>Resettable bistability:</i> When input signal is withdrawn from a bistable system, and the system resets to the low equilibrium (filled circle). <i>Hysteresis:</i> In addition to resettable bistability, when signal is increased, system goes to high equilibrium (circle). (a) Hysteresis; (b) resettable bistability, but not hysteresis. (c) No resettable bistability.</p

Comparison between human and yeast networks.

<p>(a) The best 5-node network from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006121#pcbi.1006121.g004" target="_blank">Fig 4(b)</a> that exhibits the most robust full path and partial path hysteresis. (b) Cell cycle initiation network from yeast (START network) [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006121#pcbi.1006121.ref029" target="_blank">29</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006121#pcbi.1006121.ref030" target="_blank">30</a>] exhibits robust resettable bistability in 23.81% of the full paths 12.8%. Because the networks in (a) and (b) only differ by a node with a single input and single output (<i>Myc</i>), the networks in (a) and (b) will give the same results in our analysis. (c) Best 5-node network from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006121#pcbi.1006121.g004" target="_blank">Fig 4(b)</a> with added p27 shows full path resettable bistability in 6.43% and full path hysteresis in 0.35% of the corresponding parameter paths.</p

Search for the best 5 node E2F-Rb network.

<p>(a) Full 5 node E2F-Rb network with input <i>S</i>. We test 12 subnetworks, listed in Table (b), top, for robustness of partial path and full path resettable bistability and hysteresis. (b) Each of the 12 subnetworks contains the unmarked edges in (a). In addition, we either add one of the edges 7 or 8, and/or a subset of the pair of edges (2a, 2b). The second through fifth columns list the percentage of subgraphs in <i>EPG</i>(¬<i>MD</i>) satisfying indicated query. Top three results in each column are emphasized. Note that top three networks in first three queries agree. While the very best network under full path resettable bistability is the same as for full path hyesteresis, none of the top three networks for full path hysteresis has either edge 7 or the edge 8.</p

Examples of mosquito trajectories.

<p>Example mosquito trajectories for upwind (UW), downwind (DW), and crosswind (CW) plume-finding behaviors. The stationary hosts are distributed into two groups and the contour of the odor plume marks a snapshot of the sensing threshold of the mosquito. The UW and CW mosquitoes successfully locate a host; the DW mosquito is unsuccessful. The CW mosquito leaves and re-enters the domain.</p

Growing odor plume in a large domain.

<p>Mosquitoes of all three plume-finding behaviors ( upwind, downwind, crosswind) near an evolving plume about 45–70 m long. All distances are in meters and only the outermost contour of the plume is shown. Left: The plume after 250 s. The upwind and downwind mosquitoes are already segregating into the upwind and downwind sides of the domain. Right: The same part of the domain after 500 s. The plume is about 2/3 longer and only crosswind mosquitoes remain in the vicinity of the plume.</p

Mosquito flight direction choice.

<p>Schematic of the mosquito direction choice in the host-seeking model. At each discrete flight segment, the angle represents the ideal target direction of the mosquito, which can only be sensed or followed to a precision of .</p

Ratio of contacts between a small and a large host group.

<p>Mean and standard deviation of the ratio of contacts in the small group to that in the large group (S/L) vs the ratio of group sizes. The line denotes an equal per capita contact rate between groups, while the line denotes an equal number of contacts at each group. The data corresponding to a given ratio of hosts have been separated slightly in the figure for visual purposes only.</p