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Control of complex networks requires both structure and dynamics

Alexander J. Gates, Luis M. Rocha

arXiv:1509.08409v2q-bio.MNcs.SYeess.SYmath.OC

TL;DR

The paper asks whether graph-only methods can predict control in complex systems, where control strategies matter for biological models. It tests those methods against Boolean-network dynamics and biochemical regulation models, finding that structure alone can misestimate control and that canalization matters. The authors conclude that control prediction should account for both structure and dynamics.

  • Problem

    It is unclear how well network structure predicts dynamics from the viewpoint of control, despite the need to identify control strategies for complex systems.

    Source: Introduction · p. 1

  • Method

    The authors compare graph-only control predictions with Boolean-network ensembles and biochemical regulation models, examining how transition-function canalization relates to prediction accuracy.

    Source: Introduction · p. 2, Canalization and Controllability · p. 7, Discussion · p. 7

  • Results

    Structure-only methods can underestimate or overestimate both the number and identity of driver variables, and canalization plays an important role in structure-dynamics relationships.

    Source: Discussion · pp. 7–8

  • Takeaways & Limitations

    Control-prediction methods should combine structural and dynamical properties rather than rely on network structure alone.

    Source: Discussion · p. 8

    Without more information about variable dynamics, one cannot assume a multivariate system meets the assumptions of structure-only methods.

    Source: Discussion · p. 7

Abstract

from arXiv · show

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.

Introduction

Network structure captures interactions but not the dynamics that shape control, a central concern for complex systems and biochemical regulation. This study tests how well graph-only control methods predict control in Boolean networks and biochemical models.

Introduction

  • Control strategies matter in biology because they may help steer modeled systems between diseased and healthy or mature and pluripotent states.Control predictions can also aid the separate question of how accurately a model represents the real system.Source: Introduction · p. 1
  • The central question is how well network structure predicts underlying system dynamics, especially from the viewpoint of control.Prior work links structure to some dynamical properties, while other system attributes depend on variable and interaction dynamics.Source: Introduction · p. 1
  • Graph-only structural controllability and minimum dominating set methods identify minimal driver-variable sets without considering variable dynamics.Both reduce systems to interaction graphs; structural controllability and minimum dominating sets use different assumptions about how signals propagate.Source: Introduction · pp. 1–2
  • The authors assess structure-only predictions using Boolean-network ensembles and models of segment polarity, budding-yeast cell cycle, and Arabidopsis floral organ arrangement.They aim to quantify the discrepancy between graph-only predictions and control as it unfolds in Boolean networks.Source: Introduction · p. 2
  • Structure-only predictions can both underestimate and overestimate the number and identity of necessary driver variables, so they do not realistically portray control in biochemical systems.The study compares structural controllability and minimum dominating sets with control in dynamical models.Source: Introduction · p. 2

Quantifying Control in Boolean Networks

Boolean networks represent complex systems as synchronously updated Boolean variables and capture their behavior in a state-transition graph. The control measures extend that graph with interventions and assess reachability across configurations or attractors.

Background

  • A Boolean network has Boolean-valued variables updated synchronously by deterministic logical functions, and its state-transition graph represents configurations and their transitions.Because the graph is finite, at least one attractor—a repeated configuration or cycle—must occur.Source: Background · p. 2

Control Measures

  • Control interventions are instantaneous bit flips to driver variables, adding possible transitions from each configuration to a controlled state-transition graph.The controlled graph extends the natural state-transition graph with intervention transitions.Source: Control Measures · p. 2, Figure 1 · p. 3
  • Full control requires every configuration to be reachable from every other in the controlled graph; mean configuration reachability measures the average fraction reachable from a starting configuration.Reachability is normalized by the number of other configurations.Source: Control Measures · p. 3
  • The mean controlled-configuration measure counts configurations reachable through control that were not already reachable through natural dynamics.It therefore distinguishes intervention-enabled reachability from transitions already present without control.Source: Control Measures · p. 3
  • The controlled attractor graph represents attractors as nodes and links them when a path between their basins exists, focusing on steering between attractors rather than transient configurations.This measure is relevant when attractors represent cell types or diseased and normal conditions.Source: Control Measures · p. 3

Control Portraits of Complex Systems

Control portraits show that identical network structures can yield widely different Boolean-network controllability, so structure-only predictions often miss which driver sets work. Canalization can alter that relationship: it may create mismatches, but appropriately selected canalization can also improve control and prediction.

Boolean Network Ensembles

  • The Boolean-network ensembles distinguish non-contingent functions from contingent ones, then separate contingent networks into reduced-effective-structure and full-effective-structure subsets.The full-effective-structure subset retains all underlying interactions as dynamically relevant.Source: Boolean Network Ensembles · p. 4

Network Motifs

  • For the Feed-Forward motif, structural control predicts x1 can fully control the network, but x1 fully controls only 8 of 64 Boolean networks; within the FES subset, only 4 of 20.Source: Network Motifs · p. 4
  • In the loop motif, single-variable sets fail to fully control 77% of networks, and two-variable sets fail for 44%, despite structure-only predictions of sufficiency.SC predicts any single variable suffices, while MDS predicts any two variables suffice.Source: Network Motifs · p. 5

Drosophila melanogaster

  • In the Drosophila segment-polarity model, SC’s four size-four sets do not fully control dynamics, while five other size-four sets attain greater controllability, up to CD ≈ 0.124.The SC sets reach at most RD ≈ 0.071 and vary in control despite equal support under SC.Source: Drosophila melanogaster · p. 6

Saccharomyces cerevisiae

  • In the yeast cell-cycle model, SC’s single driver achieves only about 0.19 configuration control; three size-four sets achieve full attractor control, but neither SC nor MDS predicts them.The eight MDS sets of size four do not achieve full control, and their random-dynamics control values vary substantially.Source: Saccharomyces cerevisiae · p. 6

Models of biochemical regulation

Canalization and Controllability

  • In the full-effective-connectivity yeast-structure ensemble, SC and MDS still fail, although all sampled networks are fully controlled by three drivers and many by two.The original yeast model requires four variables for full attractor control.Source: Canalization and Controllability · p. 7
  • Appropriately selected canalization can improve controllability and structure-only accuracy by reducing effective structure to linear chains, but such matches become rare in larger networks.In these cases, the effective structure better matches structure-only method assumptions.Source: Canalization and Controllability · p. 7

Discussion

Structure-only methods can both misestimate how many variables control Boolean-network dynamics and fail to identify the best driver sets. The results vary with network dynamics, including canalization, so structure alone cannot establish how a new system is controlled.

Discussion

  • Structure-only methods can over- or underestimate driver-variable needs and miss the best sets across measures of configuration access, access beyond natural dynamics, and attractor transitions.The analysis also finds substantial variation in real control among systems with the same structure and number of driver variables.Source: Discussion · p. 7
  • Canalizing transition functions generally make structure-only predictions less effective, though orchestrated canalization can align effective structure with their assumptions.Such effective structures were easy to find in small networks but rare among the dynamics of larger networks.Source: Discussion · p. 7
  • Without information about variable dynamics, one cannot assume that a system meets the assumptions of structure-only control methods.The discussion contrasts the yeast cell cycle, where canalization makes control harder than predicted, with the segment polarity network, where it facilitates control toward the wild-type attractor.Source: Discussion · pp. 7–8
  • Control and robustness are complementary: robustness measures perturbations that leave dynamics invariant, while control tracks perturbations that alter them.The paper identifies further study of controlled state-transition and controlled attractor graphs as a way to examine their relationship.Source: Discussion · p. 8
  • The study uses Boolean networks, but its control measures apply to discrete deterministic systems; identifying driver variables is distinct from constructing intervention sequences.The authors expect a greater structure–dynamics disparity when the goal is to identify specific controllers.Source: Discussion · p. 8
  • Future control-prediction methods should combine structural and dynamical properties while accounting for computational complexity.The discussion names monotone control systems, master stability functions, schema redescription, and stabilization subgraphs as promising approaches.Source: Discussion · p. 8

Additional information

The additional information includes captions for control visualizations of Boolean-network ensembles and two biochemical-regulation models, plus a competing-interests declaration.

Additional information

  • The authors declare no competing financial interests.Source: Additional information · p. 11
  • Figure 4 describes control portraits for a Boolean-network ensemble constrained by the Loop motif with self-interactions.The caption identifies reachable configurations and, in the fuller description, reachable attractors and controllable configurations across ensemble subsets.Source: Figure 4 · p. 13, Additional information · p. 13
  • Figure 5 depicts segment-polarity-network control across driver sets of sizes one through four, highlighting structural-controllability predictions and a full-attractor-control set.The inset reports mean reachable attractors for singleton driver sets.Source: Figure 5 · p. 14
  • Figure 6 depicts budding-yeast cell-cycle control across driver sets of sizes one through four and controlled attractor graphs for singleton sets.Its caption distinguishes structural-controllability and minimum-dominating-set predictions from full-attractor-control sets, and marks the wild-type attractor separately.Source: Figure 6 · p. 15
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