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Efficient parameter search for qualitative models of regulatory networks using symbolic model checking

Grégory Batt, Michel Page, Irene Cantone, Gregor Goessler, Pedro T. Monteiro, Hidde De Jong

arXiv:1005.2107v1q-bio.QM

TL;DR

Regulatory-network analysis asks whether structure accounts for observed behavior or can produce desired dynamics, but searching qualitative-model parameter spaces can be difficult. This paper uses symbolic model checking to avoid exhaustive parameter enumeration and applies it to IRMA, finding biologically interpretable compatible constraints and suggesting rewiring for a more robust galactose response.

  • Problem

    Testing whether regulatory-network structures match observed behavior or produce desired behavior requires parameter search, which becomes difficult when state and parameter spaces are too large for exhaustive testing.

    Source: §1 · p. 5

  • Method

    The method symbolically encodes qualitative PADE model structure, parameter constraints, and transition rules for temporal-logic model checking without exhaustive parameter enumeration.

    Source: §1 · p. 5, §6 · p. 19

  • Results

    For IRMA, compatible parameter constraints are precise, biologically interpretable, and found within minutes even for complex dynamical properties; the results suggest effective rewiring is needed for a more robust galactose response.

    Source: §1 · p. 6, §6 · p. 19

  • Takeaways & Limitations

    The approach tests whether a regulatory-network structure is consistent with observed behavior or can generate a desired behavior, and the IRMA results suggest that more robust galactose control may require network rewiring.

    Source: §6 · pp. 18–19

    The PADE model assumes qualitative dynamics are tied to network structure and regulatory logic independently of kinetic details, an approximation whose validity requires care; the paper proposes extending the search to incompletely specified regulation functions.

    Source: §2.3 · p. 9, §6 · p. 19

Abstract

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Investigating the relation between the structure and behavior of complex biological networks often involves posing the following two questions: Is a hypothesized structure of a regulatory network consistent with the observed behavior? And can a proposed structure generate a desired behavior? Answering these questions presupposes that we are able to test the compatibility of network structure and behavior. We cast these questions into a parameter search problem for qualitative models of regulatory networks, in particular piecewise-affine differential equation models. We develop a method based on symbolic model checking that avoids enumerating all possible parametrizations, and show that this method performs well on real biological problems, using the IRMA synthetic network and benchmark experimental data sets. We test the consistency between the IRMA network structure and the time-series data, and search for parameter modifications that would improve the robustness of the external control of the system behavior.

Efficient Parameter Search

The paper frames structure–behavior questions for regulatory networks as parameter search in qualitative models. It develops symbolic model checking to avoid enumerating possible parameterizations and applies the approach to IRMA and galactose-response robustness.

Paramétrisation efficace de modèles qualitatifs de réseaux de régulation à l’aide de model checking symbolique

  • The motivating questions are whether a proposed network structure fits observed behavior and whether it can generate a desired behavior.The paper casts both questions as parameter search for qualitative regulatory-network models.Source: Paramétrisation efficace de modèles… · p. 4
  • Symbolic model checking avoids enumerating all possible parameterizations in the qualitative-model search.The method is applied to the synthetic IRMA network and associated benchmark experimental data.Source: Paramétrisation efficace de modèles… · p. 4
  • The study tests IRMA’s consistency with time-series data and searches for parameter changes that could make galactose control more robust.Source: Paramétrisation efficace de modèles… · p. 4

1 Introduction

The paper addresses how regulatory-network structure relates to dynamics, including whether a proposed structure matches observations or can generate a desired behavior. It uses symbolic model checking to search qualitative-model parameters and reports an IRMA application with compatible parameter values and proposed network rewiring.

1 Introduction

  • Qualitative models enable structure–dynamics analysis through finite parameter searches and suit biological data that are often semi-quantitative.They emphasize qualitative trends rather than precise quantitative values.Source: §1 · p. 5
  • Large qualitative models remain difficult to search because their state and parameter spaces can exceed what existing methods can test exhaustively.The paper formulates dynamic properties in temporal logic and checks them with a model checker.Source: §1 · p. 5
  • Symbolically encoding model structure, parameter constraints, and transition rules supports efficient analysis of large state spaces without enumerating parameterizations.This extends the authors’ previous work.Source: §1 · p. 5
  • Compared with related work, the method handles incompletely parametrized models, gives more precise results, and avoids strongly simplifying PADE dynamics.Source: §1 · p. 6
  • For IRMA, the method finds parameter values compatible with individual and averaged expression profiles, with only a small part of parameter space fitting observations.The identified constraints are biologically relevant, and the authors propose network rewiring to improve robustness of external control.Source: §1 · p. 6

2 Qualitative model of IRMA network

The section introduces IRMA, its measured switch-on and switch-off dynamics, and a PADE model of its regulatory logic. It also states the model’s assumptions and motivates testing data consistency and more robust galactose response.

2.1 IRMA network

  • IRMA is a synthetic yeast network built from five well-characterized genes and designed to reduce crosstalk with the host; its structure includes positive and negative feedback loops.These feedbacks suggest that the network might act as a synthetic oscillator for suitable parameter values.Source: §2.1 · p. 6

2.2 Measurements of IRMA dynamics

  • Switch-off experiments shut down transcription of all genes, while averaged switch-on profiles appear oscillatory, with Swi5 peaks at 40 and 180 minutes.The averaged profiles combine five switch-on and four switch-off experiments, and the large unshown confidence intervals make the data essentially semi-quantitative.Source: §2.2 · p. 7
  • Individual switch-on time-series do not all show oscillatory behavior, so average expression can misrepresent network behavior.The profiles were measured by qRT-PCR every 10 minutes for switch-off or 20 minutes for switch-on over more than three hours.Source: §2.2 · p. 7

2.3 PADE model of IRMA network

  • The PADE model represents regulatory synthesis with step-function logic and protein or RNA concentrations, alongside first-order degradation.Synthesis constants are modulated by regulation functions; thresholds determine the step functions, and degradation constants include growth dilution and protein degradation.Source: §2.3 · pp. 7, 9
  • For IRMA, the model uses five protein-concentration variables and a galactose input, while assuming mRNA and protein variations are the same.The model encodes regulation such as Gal4 activation of SWI5 only when Gal80 is absent or galactose is present, and assumes Ash1 inhibition overrides Swi5 activation.Source: §2.3 · p. 9
  • The PADE translation approximates Hill-type rates with step functions and assumes qualitative dynamics depend on network structure and regulatory logic rather than kinetic details.The authors note that care is needed in deciding when this modeling approximation is valid; they seek consistency with observed trends and parameter changes for more robust galactose response.Source: §2.3 · pp. 9–10

3 Search of parameter space using symbolic model checking

The section builds a qualitative state-transition representation of PADE dynamics and uses symbolic model checking to search parameter orderings without enumerating every model. Its encoding discretizes domains and expresses transitions and temporal-logic properties in a form the model checker can analyze.

3.1 Qualitative analysis of PADE models

  • The state space is partitioned at threshold and focal values so each domain has a uniform derivative-sign pattern and can serve as a state in a transition graph.Edges represent trajectories that move between domains without leaving their union.Source: §3.1 · pp. 10–11
  • A parameterization is the total order of threshold and focal parameters; the qualitative dynamics remain unchanged within regions sharing that order.Thus exact parameter values are unnecessary for determining the state-transition graph.Source: §3.1 · pp. 11–12

3.2 Search of parameter space: a model-checking approach

  • Symbolic model checking avoids enumerating every parameterization and generating its full transition graph, which becomes infeasible as network size grows.Observed or desired trends are encoded as temporal-logic properties and checked against the graph.Source: §3.2 · p. 12

3.3 Symbolic encoding of PADE model and dynamics

  • The discretization function maps each domain to integer coordinates, using even coordinates on threshold or focal planes and odd coordinates elsewhere.These coordinates encode domains and neighboring states for the symbolic representation.Source: §3.3 · p. 12
  • An interval-based expression computes the flow sign used to determine transitions, including at threshold planes where enumerating surrounding domains would be exponential.The symbolic encoding constrains parameters to remain fixed, domains to be neighbors, and their relative positions to satisfy transition conditions; temporal-logic checks can then return a valid parameterization.Source: §3.3 · p. 13

4 Validation: consistency of IRMA network with experimental data

The authors encode qualitative trends from averaged and individual IRMA time series as temporal-logic properties and test whether parameterizations satisfy them. The data are selective: only a small subset of the enumerated parameterizations is compatible, and the resulting constraints yield specific regulatory insights.

4.1 Temporal-logic encoding of observations

  • The individual experiments differ qualitatively, so the authors define φ2 to require a path for each of the 9 observed switch-on and switch-off behaviors.The specification uses trends only and includes nearly 160 derivative-sign constraints; changes below 5·10^-3 units are treated as insignificant, and measurements immediately before the medium shift are excluded.Source: §4.1 · p. 14

4.2 Testing consistency of network with observations

  • A parameterization satisfying the averaged data was found in 49 seconds, and a counterexample to ¬φ1 returned one in 100 seconds for a state space of nearly 50,000 states.The tested parameter space contained nearly 5,000 orderings.Source: §4.2 · p. 15
  • One compatible parameterization places Gal80's threshold above its upper focal parameter, suggesting switch-off may occur without Gal80 inhibition and independently of galactose.For this parameterization, GNA also identifies a stable low-expression switch-off state and oscillation-consistent strongly connected components; the attractors coexist with or without galactose.Source: §4.2 · p. 15
  • The averaged-data parameterization does not satisfy φ2, but another parameterization satisfies the individual-data property in 237 seconds.In that parameterization, all thresholds lie between basal and upregulated focal parameters.Source: §4.2 · p. 15

4.3 Detailed analysis of valid parameter set

  • 12 of 4860 parameterizations satisfy the averaged time-series property, while only 4 satisfy the individual time-series property.The authors conclude that the properties extracted from the data are selective.Source: Table 1 · p. 16, §4.3 · p. 17
  • Compatible parameterizations require Swi5 to activate CBF1 at a lower threshold than it activates ASH1 and GAL80, consistent with independent promoter-activity evidence.They also permit effective regulation of GAL80 by Swi5 to be unimportant, consistent with a reported 6-fold SWI5 increase producing only a 1.08-fold GAL80 increase.Source: §4.3 · p. 17

5 Re-engineering: improving external control by galactose

The authors specify a desired switch-off state without galactose and oscillatory Swi5 behavior with galactose, while requiring galactose to drive the system out of the low-Swi5 state. Symbolic search finds parameterizations meeting this objective, and the resulting constraints suggest a simpler network design with two feedback loops removed.

5.1 Temporal-logic specification of design objective

  • The design objective requires low Swi5 eventually without galactose, oscillatory Swi5 with galactose, and departure from low Swi5 under galactose.The specification combines reachability and necessity conditions across initial states.Source: §5.1 · pp. 17–18

5.2 Parametrizations consistent with design objective

  • For a proposed parameterization, galactose permits oscillations from the switch-off state, whereas without galactose the all-genes-off steady state attracts the entire state space.The galactose condition has two terminal strongly connected components attracting much of the state space.Source: §5.2 · p. 18
  • The symbolic search finds parameterizations satisfying the design property in about 2 minutes, although none of the data-consistent parameterizations satisfies it.Exhaustive enumeration finds only 7 valid parameterizations among 4860, taking 1300 seconds, compared with 126 seconds for the method's parameter ordering search.Source: §5.2 · p. 18
  • The design constraints require Swi5 to activate CBF1 consistently, suggesting removal of Swi5 regulation of CBF1.The authors explain that intermittent Swi5 could otherwise disrupt CBF1 and Gal4 expression and allow convergence to switch-off.Source: §5.2 · p. 18
  • The constraints also make Swi5 regulation of GAL80 ineffective, yielding a simplified design in which two of the three feedback loops are removed.The threshold condition means either the GAL80 promoter is always activated or Gal80 is always sufficient to repress SWI5.Source: §5.2 · p. 18

6 Discussion

The paper presents symbolic model checking as an efficient way to search qualitative regulatory-network parameter spaces and test whether structures can produce observed or desired behaviors. Its IRMA application yields biologically interpretable constraints consistent with independent observations, while further work could allow regulation functions as well as parameters to be unspecified.

  • The method searches parameter spaces of qualitative regulatory-network models to test consistency with observed behavior or the ability to generate desired behavior.Source: §6 · p. 18
  • Symbolic encoding avoids explicit state-space generation and enumeration of possible parametrizations, enabling efficient model checking of incompletely parametrized PADE models.The main ideas also carry over to logical models.Source: §6 · pp. 18–19
  • For IRMA, the method yields precise, biologically interpretable parameter constraints consistent with independent experimental observations, with complex-property searches taking at most a few minutes.The results seem to confirm IRMA's intended separation from the host network and suggest effective rewiring for a more robust response to galactose.Source: §6 · p. 19
  • Unlike quantitative ODE models that single out one likely behavior, the approach scans the entire parameter space and produces general results independent of specific molecular representations and parameter values.The paper characterizes qualitative and quantitative approaches as complementary.Source: §6 · p. 19
  • Compared with related verification methods, the approach handles incompletely parametrized models and temporal-logic properties while representing threshold dynamics at a finer abstraction level suited to IRMA time-series comparison.Source: §6 · p. 19
  • A proposed direction for further research is to search models in which regulation functions, as well as parameters, are incompletely specified.Source: §6 · p. 19

A Transition rules

The appendix gives transition rules for moving between domains in the PADE state-space partition. The rules distinguish transitions that remain in a domain, enter a higher-dimensional domain, or enter a lower-dimensional one.

A Transition rules

  • For an internal transition, where D = D′, every coordinate at a threshold or focal value must permit zero derivative for a solution to remain in the domain.The proof explains that without this condition, solutions instantaneously leave the domain.Source: A Transition rules · p. 20
  • A dimension-increasing transition requires zero derivative for coordinates on threshold or focal values and a positive feasible flow in each changing coordinate consistent with the domains’ relative positions.These conditions encode whether solutions can remain on the relevant planes and move in the required direction.Source: A Transition rules · p. 20
  • A dimension-decreasing transition requires zero derivative for coordinates shared on threshold or focal values and positive feasible flow in each coordinate that changes between domains.The appendix states the transition criteria and sketches sufficiency under these conditions or an alternative condition.Source: A Transition rules · p. 21

B Comparison with previous definition of dynamics

The paper’s proposed dynamics definition is more conservative than an earlier definition based on neighboring regular domains. The resulting set-valued dynamics is usually equal to the earlier one, with strict inclusion confined to two dual-role cases.

B Comparison with previous definition of dynamics

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