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Control of complex networks requires both structure and dynamics.

Gates AJ, Rocha LM - Sci Rep (2016)

Bottom Line: Two recent methodologies suggest that the controllability of complex systems can be predicted solely from the graph of interactions between variables, without considering their dynamics: structural controllability and minimum dominating sets.We study Boolean network ensembles of network motifs as well as three models of biochemical regulation: the segment polarity network in Drosophila melanogaster, the cell cycle of budding yeast Saccharomyces cerevisiae, and the floral organ arrangement in Arabidopsis thaliana.Our analysis further shows that the logic of automata transition functions, namely how canalizing they are, plays an important role in the extent to which structure predicts dynamics.

View Article: PubMed Central - PubMed

Affiliation: School of Informatics and Computing, Indiana University, Bloomington, IN, USA.

ABSTRACT
The study of network structure has uncovered signatures of the organization of complex systems. However, there is also a need to understand how to control them; for example, identifying strategies to revert a diseased cell to a healthy state, or a mature cell to a pluripotent state. Two recent methodologies suggest that the controllability of complex systems can be predicted solely from the graph of interactions between variables, without considering their dynamics: structural controllability and minimum dominating sets. We demonstrate that such structure-only methods fail to characterize controllability when dynamics are introduced. We study Boolean network ensembles of network motifs as well as three models of biochemical regulation: the segment polarity network in Drosophila melanogaster, the cell cycle of budding yeast Saccharomyces cerevisiae, and the floral organ arrangement in Arabidopsis thaliana. We demonstrate that structure-only methods both undershoot and overshoot the number and which sets of critical variables best control the dynamics of these models, highlighting the importance of the actual system dynamics in determining control. Our analysis further shows that the logic of automata transition functions, namely how canalizing they are, plays an important role in the extent to which structure predicts dynamics.

No MeSH data available.


Control portrait of the BN ensemble constrained by the Loop network motif with self-interactions.(A) The mean fraction of reachable configurations  and the mean fraction of controllable configurations  for the full ensemble of 4096 BNs with structure given by the Loop network motif with self-interactions shown in Fig. 1D, as controlled by the driver variable sets  (STG), D ≡ {xi} and D ≡ {xi, xj} (due to the symmetry of the network, all sets of size one are equivalent, likewise those of size two). The full effective structure (FES) subset is shown by red circles, the reduced effective structure (RES) subset is shown in blue squares, and the non-contingent (NC) subset is shown by green diamonds; the area of the object corresponds to the number of networks at that point. (B) (left) The number of attractors for each network in the full ensemble spans from 1–8, the area of each pie chart scales logarithmically with the number of attractors, from 1868 to 1; the colored slices delineate the subset decompositions for NC, RES, and FES. (middle and right) Box plots for the distribution of the mean fraction of reachable attractors  for D ≡ {xi}, {xi, xj} for the full ensemble (purple), NC, RES, and FES subsets. In each case, the box shows the interquartile range, the median is given by the solid vertical line, the mean is given by the black circle, and the whiskers show the support of the distribution.
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f4: Control portrait of the BN ensemble constrained by the Loop network motif with self-interactions.(A) The mean fraction of reachable configurations and the mean fraction of controllable configurations for the full ensemble of 4096 BNs with structure given by the Loop network motif with self-interactions shown in Fig. 1D, as controlled by the driver variable sets (STG), D ≡ {xi} and D ≡ {xi, xj} (due to the symmetry of the network, all sets of size one are equivalent, likewise those of size two). The full effective structure (FES) subset is shown by red circles, the reduced effective structure (RES) subset is shown in blue squares, and the non-contingent (NC) subset is shown by green diamonds; the area of the object corresponds to the number of networks at that point. (B) (left) The number of attractors for each network in the full ensemble spans from 1–8, the area of each pie chart scales logarithmically with the number of attractors, from 1868 to 1; the colored slices delineate the subset decompositions for NC, RES, and FES. (middle and right) Box plots for the distribution of the mean fraction of reachable attractors for D ≡ {xi}, {xi, xj} for the full ensemble (purple), NC, RES, and FES subsets. In each case, the box shows the interquartile range, the median is given by the solid vertical line, the mean is given by the black circle, and the whiskers show the support of the distribution.

Mentions: Let us now consider the N = 3 variable loop motif with self-interactions (Fig. 2D). The full ensemble of BNs constrained by this motif is much larger than the previous example (every variable has ki = 2 inputs); it consists of 4096 networks of which 1352 are NC, 1744 have RES, and 1000 have FES. Figure 4A shows the control portrait of this motif’s BN ensemble for a single (D ≡ {xi}) or pair (D ≡ {xi, xj}) of driver variables. The control portrait of the STG illustrates the difference between the two measures of controllability. While varies greatly, for all BNs. This means that in some BNs, many configurations can be reached simply because the transient dynamics move through many network configurations. Structural control methodologies ignore this natural propensity for control (self-organization). Thus we use the measure to tally only the proportion of transitions that result from control interventions.


Control of complex networks requires both structure and dynamics.

Gates AJ, Rocha LM - Sci Rep (2016)

Control portrait of the BN ensemble constrained by the Loop network motif with self-interactions.(A) The mean fraction of reachable configurations  and the mean fraction of controllable configurations  for the full ensemble of 4096 BNs with structure given by the Loop network motif with self-interactions shown in Fig. 1D, as controlled by the driver variable sets  (STG), D ≡ {xi} and D ≡ {xi, xj} (due to the symmetry of the network, all sets of size one are equivalent, likewise those of size two). The full effective structure (FES) subset is shown by red circles, the reduced effective structure (RES) subset is shown in blue squares, and the non-contingent (NC) subset is shown by green diamonds; the area of the object corresponds to the number of networks at that point. (B) (left) The number of attractors for each network in the full ensemble spans from 1–8, the area of each pie chart scales logarithmically with the number of attractors, from 1868 to 1; the colored slices delineate the subset decompositions for NC, RES, and FES. (middle and right) Box plots for the distribution of the mean fraction of reachable attractors  for D ≡ {xi}, {xi, xj} for the full ensemble (purple), NC, RES, and FES subsets. In each case, the box shows the interquartile range, the median is given by the solid vertical line, the mean is given by the black circle, and the whiskers show the support of the distribution.
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Related In: Results  -  Collection

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Show All Figures
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f4: Control portrait of the BN ensemble constrained by the Loop network motif with self-interactions.(A) The mean fraction of reachable configurations and the mean fraction of controllable configurations for the full ensemble of 4096 BNs with structure given by the Loop network motif with self-interactions shown in Fig. 1D, as controlled by the driver variable sets (STG), D ≡ {xi} and D ≡ {xi, xj} (due to the symmetry of the network, all sets of size one are equivalent, likewise those of size two). The full effective structure (FES) subset is shown by red circles, the reduced effective structure (RES) subset is shown in blue squares, and the non-contingent (NC) subset is shown by green diamonds; the area of the object corresponds to the number of networks at that point. (B) (left) The number of attractors for each network in the full ensemble spans from 1–8, the area of each pie chart scales logarithmically with the number of attractors, from 1868 to 1; the colored slices delineate the subset decompositions for NC, RES, and FES. (middle and right) Box plots for the distribution of the mean fraction of reachable attractors for D ≡ {xi}, {xi, xj} for the full ensemble (purple), NC, RES, and FES subsets. In each case, the box shows the interquartile range, the median is given by the solid vertical line, the mean is given by the black circle, and the whiskers show the support of the distribution.
Mentions: Let us now consider the N = 3 variable loop motif with self-interactions (Fig. 2D). The full ensemble of BNs constrained by this motif is much larger than the previous example (every variable has ki = 2 inputs); it consists of 4096 networks of which 1352 are NC, 1744 have RES, and 1000 have FES. Figure 4A shows the control portrait of this motif’s BN ensemble for a single (D ≡ {xi}) or pair (D ≡ {xi, xj}) of driver variables. The control portrait of the STG illustrates the difference between the two measures of controllability. While varies greatly, for all BNs. This means that in some BNs, many configurations can be reached simply because the transient dynamics move through many network configurations. Structural control methodologies ignore this natural propensity for control (self-organization). Thus we use the measure to tally only the proportion of transitions that result from control interventions.

Bottom Line: Two recent methodologies suggest that the controllability of complex systems can be predicted solely from the graph of interactions between variables, without considering their dynamics: structural controllability and minimum dominating sets.We study Boolean network ensembles of network motifs as well as three models of biochemical regulation: the segment polarity network in Drosophila melanogaster, the cell cycle of budding yeast Saccharomyces cerevisiae, and the floral organ arrangement in Arabidopsis thaliana.Our analysis further shows that the logic of automata transition functions, namely how canalizing they are, plays an important role in the extent to which structure predicts dynamics.

View Article: PubMed Central - PubMed

Affiliation: School of Informatics and Computing, Indiana University, Bloomington, IN, USA.

ABSTRACT
The study of network structure has uncovered signatures of the organization of complex systems. However, there is also a need to understand how to control them; for example, identifying strategies to revert a diseased cell to a healthy state, or a mature cell to a pluripotent state. Two recent methodologies suggest that the controllability of complex systems can be predicted solely from the graph of interactions between variables, without considering their dynamics: structural controllability and minimum dominating sets. We demonstrate that such structure-only methods fail to characterize controllability when dynamics are introduced. We study Boolean network ensembles of network motifs as well as three models of biochemical regulation: the segment polarity network in Drosophila melanogaster, the cell cycle of budding yeast Saccharomyces cerevisiae, and the floral organ arrangement in Arabidopsis thaliana. We demonstrate that structure-only methods both undershoot and overshoot the number and which sets of critical variables best control the dynamics of these models, highlighting the importance of the actual system dynamics in determining control. Our analysis further shows that the logic of automata transition functions, namely how canalizing they are, plays an important role in the extent to which structure predicts dynamics.

No MeSH data available.