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Perturbation Biology: inferring signaling networks in cellular systems

Evan J. Molinelli, Anil Korkut, Weiqing Wang, Martin L. Miller, Nicholas P. Gauthier, Xiaohong Jing, Poorvi Kaushik, Qin He, Gordon Mills, David B. Solit, Christine A. Pratilas, Martin Weigt, Alfredo Braunstein, Andrea Pagnani, Riccardo Zecchina, Chris Sander

arXiv:1308.5193v1q-bio.MN

TL;DR

Cancer therapy design needs quantitatively predictive models of context-specific signaling interactions. The paper combines systematic drug perturbations and proteomic or phenotypic measurements with de novo nonlinear network inference using Belief Propagation. The resulting melanoma models reproduce known pathway biology, predict novel interactions and drug responses, and support experimentally tested therapeutic predictions.

  • Problem

    Cancer drug design is hindered by incomplete quantitative understanding of coordinated signaling-protein interactions in specific cellular contexts.

  • Method

    The method infers de novo nonlinear signaling networks from systematic single and paired drug perturbations using Belief Propagation.

  • Results

    In RAF-inhibitor-resistant SKMEL-133 melanoma, the models agree with known biology, predict novel interactions and efficacious targets, and predict responses to untested drug combinations.

  • Takeaways & Limitations

    The technology provides a proof-of-principle framework for using predictive network models to explore drug combinations and context-specific therapeutic strategies.

  • Takeaways & Limitations

    BP can confuse correlation with causation from steady-state data, and its inferred solutions depend on discretization and initialization.

Abstract

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We present a new experimental-computational technology of inferring network models that predict the response of cells to perturbations and that may be useful in the design of combinatorial therapy against cancer. The experiments are systematic series of perturbations of cancer cell lines by targeted drugs, singly or in combination. The response to perturbation is measured in terms of levels of proteins and phospho-proteins and of cellular phenotype such as viability. Computational network models are derived de novo, i.e., without prior knowledge of signaling pathways, and are based on simple non-linear differential equations. The prohibitively large solution space of all possible network models is explored efficiently using a probabilistic algorithm, belief propagation, which is three orders of magnitude more efficient than Monte Carlo methods. Explicit executable models are derived for a set of perturbation experiments in Skmel-133 melanoma cell lines, which are resistant to the therapeutically important inhibition of Raf kinase. The resulting network models reproduce and extend known pathway biology. They can be applied to discover new molecular interactions and to predict the effect of novel drug perturbations, one of which is verified experimentally. The technology is suitable for application to larger systems in diverse areas of molecular biology.

Author Summary

The paper presents de novo network inference for context-specific cancer signaling, using perturbation data and Belief Propagation to model and predict drug responses.

  • Author Summary: Belief Propagation constructs predictive mathematical models of signaling responses from experimental perturbation data.The approach avoids dependence on prior pathway knowledge and explores highly probable network models.
  • Author Summary: Applied to a BRAF-mutant melanoma cell line, the method agrees with known biology and predicts novel interactions and efficacious drug targets.

Introduction

The paper develops context-specific, de novo signaling models from systematic drug perturbations, addressing the combinatorial difficulty of network inference with Belief Propagation.

  • Introduction: The method constructs context-specific predictive networks without relying on potentially inappropriate prior molecular interactions.This is motivated by genetic and proteomic differences across cancer contexts.
  • Introduction: Belief Propagation addresses the combinatorial explosion of possible networks that limits Monte Carlo methods to smaller systems.
  • Introduction: Systematic paired drug perturbations, proteomic measurements, and phenotypic readouts provide data for quantitative network inference.
  • Introduction: Belief Propagation improves computational efficiency over Monte Carlo simulations without sacrificing accuracy on biologically inspired toy models.
  • Introduction: In RAF-inhibitor-resistant SKMEL-133 melanoma cells, inferred models predict proteomic and phenotypic responses to untested drug combinations.

Mathematical form of the network model

The network model represents cellular variables with coupled nonlinear differential equations driven by interactions and drug perturbations, while inference uses steady-state measurements.

  • Mathematical form of the network model: The model describes cellular dynamics with coupled nonlinear differential equations whose trajectories settle to steady states.
  • Mathematical form of the network model: Each network node represents a molecular or phenotypic quantity, expressed as a log2 ratio relative to the unperturbed condition.Positive and negative values denote increases and decreases, respectively.
  • Mathematical form of the network model: The model includes a sigmoidal nonlinearity, restoration-rate parameters, and saturation parameters to constrain variable dynamics.
  • Mathematical form of the network model: Drug perturbations are represented as external inputs to targeted nodes, and drug combinations are modeled by summing their perturbation vectors.
  • Mathematical form of the network model: Directed interaction weights quantify how upstream nodes influence the rate of change of downstream nodes.Nonzero weights define interactions in the inferred network.
  • Mathematical form of the network model: Although simulations generate trajectories, parameter inference in this work uses steady-state data, leaving simulated trajectories unconstrained by measurements.

The problem of model inference

Model inference seeks low-cost interaction matrices that reproduce experimental observations while remaining sparse enough to retain predictive power.

  • The problem of model inference: Model inference optimizes interaction-matrix parameters to balance agreement with experimental data and model complexity.
  • The problem of model inference: Lower-cost models are favored because predictive power requires both low error and low complexity.
  • The problem of model inference: The cost function combines data error with an L0 penalty on nonzero interactions to discourage overfitting and reflect network sparsity.

De novo network inference is a hard problem

De novo network inference requires searching an enormous space of possible parameter configurations, making exhaustive evaluation impractical even for moderate systems. Monte Carlo exploration helps but becomes prohibitively expensive as network size grows.

  • For N=20 nodes, the number of distinct solutions is on the order of 10^190, making explicit enumeration prohibitive.
  • Monte Carlo search explores configurations by accepting or rejecting random moves according to their cost.
  • Monte Carlo searches become prohibitively expensive for larger systems despite reasonable exploration of smaller networks.

Fast inference via a probability model of network configurations

The method replaces direct traversal of the exponentially large model space with a probabilistic formulation in which low-cost models receive higher probability. Belief propagation then efficiently approximates parameter marginals, reducing computational effort and enabling larger inference problems.

  • Models with lower cost receive higher probability, allowing inference to focus on probable network configurations rather than enumerate every model.
  • Belief propagation approximates marginal distributions in factor graphs through iterative local updates informed by global parameter distributions.
  • Belief propagation is exact on tree-shaped factor graphs, while the method here uses it as an approximate inference procedure for broader graphical models.
  • The method reduces computational complexity, substantially lowering effort for smaller systems and enabling inference problems that would otherwise be prohibitive at larger scales.
  • The iteration updates each cavity parameter distribution using experimental fitness and the distributions of non-cavity parameters, then feeds the update into subsequent steps.

Simplified probability model of network configurations

The probability model is simplified by replacing full configurations with tractable distributions over individual parameters and by decoupling node calculations at steady state. These approximations enable efficient belief-propagation updates while retaining probabilistic model selection.

  • The method reduces whole-configuration probabilities to individual parameter marginals, whose samples efficiently explore high-probability model configurations.
  • Assumption 1: Discrete set of real valued parameter assignments: Using 11 discrete parameter values centered at zero rarely failed to converge, whereas 3 values often failed and produced overly certain distributions.
  • Continuous parameter solutions are refined after inference with a local gradient-descent optimization algorithm.
  • Assumption 2: Decoupling of nodes at steady state: At steady state, substituting experimentally observed variables decouples model predictions so each row of the interaction matrix can be assigned an independent posterior distribution.
  • Belief propagation algorithm: iterative updates of probability estimates: Belief propagation updates one parameter distribution at a time inside a cavity, combining experimental fitness with evolving global information from other parameters.
  • Assumption 3: Independent model parameter distributions: The local update weights each candidate parameter by model fitness and the probability of compatible configurations, then uses factorized approximations to avoid brute-force enumeration.
  • Assumption 4: Gaussian mean-field approximation: A Gaussian mean-field approximation replaces sums over exponentially many configurations with an integral over a single scalar aggregate variable.

Network model instantiation by 'decimation' guided by belief propagation

Belief propagation produces parameter distributions rather than individual networks, so a guided decimation procedure samples those distributions to instantiate distinct model solutions. The inferred interactions remain constrained by measurement coverage.

  • Guided decimation first computes BP probabilities, then probabilistically selects an interaction and value according to those probabilities to generate distinct solutions.
  • The resulting model's direct interactions need not be direct biological interactions because unmeasured proteins and phosphoprotein states may omit intermediate players.

Technical performance

Belief propagation (BP) inferred signaling networks accurately and substantially faster than Monte Carlo (MC), including under sparse perturbations and realistic noise. Its main technical limitation was difficulty distinguishing correlation from causation in steady-state data, although perturbing relevant nodes improved causal inference.

  • Belief propagation versus Monte Carlo: R=0.7 correlation between BP-inferred and true nonzero parameters demonstrated strong agreement with only M=N training patterns.The correlation approached 1 as the number and quality of training patterns increased.
  • Belief propagation versus Monte Carlo: BP offered a tremendous speed advantage over MC at no observable loss of accuracy.MC was consistently about three orders of magnitude slower, while BP performed no worse on the tested datasets.
  • Experimental constraints: Rich perturbation data enabled near-perfect inference in toy systems, whereas biological experiments constrain perturbation coverage, reporters, drug combinations, cost, and measurement accuracy.The authors note that causal inference improved when potential interacting nodes received sufficient perturbation.
  • Interaction recovery: BP recovered 74% of true interactions and achieved 55% precision, with most errors associated with compensatory motifs or probable non-interactions.BP missed 17 of 60 true interactions; 29 of 37 false positives were structurally supported or had a significant probability of being zero.
  • Limitations: Steady-state data made it difficult for BP to disambiguate correlation from causation, because highly correlated but noncausal nodes could receive inferred interactions.The authors identify assumptions in the BP likelihood, especially decoupling from system dynamics, as a likely source of incorrect edge predictions.
  • Robustness to noise: BP remained able to identify key regulatory influences from noisy steady-state data with sparse perturbations and no prior knowledge.Recall and precision decreased with added noise, but performance approached the noise-free case when sufficient replicates were used.

Network models of signaling pathways in melanoma cells

The study infers context-specific signaling networks in SKMEL-133 melanoma cells from systematic drug perturbations, using probabilistic modeling to predict responses and recover pathway biology. The models reproduce known MAPK and PI3K/AKT interactions, suggest additional interactions and targets, and support experimentally validated predictions.

  • Network inference: Probabilistic belief propagation and nonlinear differential-equation models were used to infer executable signaling networks from perturbation-constrained data.The models included protein, phospho-protein, activity, and viability nodes, and withheld-condition responses were generated by numerical simulation.
  • Experimental design: 44 single or paired perturbations of eight targeted drugs generated proteomic and viability responses for network inference in SKMEL-133 cells.Responses comprised 16 protein or phospho-protein readouts and cell viability measurements relative to untreated cells.
  • Predictive validation: The inferred network ensemble predicted withheld proteomic and phenotypic responses with Pearson correlation 0.87 and cross-validation error CV=0.05.Leave-7-out validation inferred models from partial data and simulated the omitted perturbation conditions.
  • Network interpretation: Known RAF/MEK/MAPK and PI3K/AKT pathway interactions were recovered, while the average model also suggested additional biochemical interactions for experimental testing.The inferred models captured major canonical pathway features, including interactions involving PI3K, AKT, TSC2, mTOR, RAF, MEK, and cell-cycle proteins.

Discussion

The paper combines perturbation experiments with belief-propagation network inference to construct predictive, context-specific signaling models. In melanoma cells, the models recover known pathway interactions, suggest additional network structure, support large-scale drug-combination screening, and expose limitations tied to data quality, model assumptions, and drug specificity.

  • Network pharmacology in the era of personalized medicine: The approach addresses cross-pathway coupling, multigenic disease, and individual therapy-response variation that challenge gene-by-gene molecular biology.The paper frames network pharmacology as a combination of inference and perturbation technologies for deriving and testing therapeutic combinations.
  • Power and limitations of the belief propagation method for network modeling of signaling pathways: Belief propagation enables larger de novo signaling models than Monte Carlo search, with favorable scaling and no measurable accuracy loss in biologically realistic toy data.Under noisy sparse perturbations, BP recovered most dominant interactions, although some false positives remained.
  • Power and limitations of the belief propagation method for network modeling of signaling pathways: Model reliability depends on applicable equations and rich, high-quality perturbation data, while drug off-target effects can tether predictions to the inhibitor used during training.The authors identify improved time-dependent inference, incorporation of prior knowledge, and richer phenotypic measurements as method extensions.
  • Confirmed and predicted interactions in malignant melanoma cells: De novo modeling recovered known MAPK and AKT/PI3K interactions and predicted a bidirectional RB–MAPK interaction plus a PI3K/MAPK feedback loop involving Src.The predicted interactions extend the canonical pathway structure captured by the inferred models.
  • Predicting effective novel drug combinations with BP based network modeling: Predictive network models can expand drug-combination searches from a few hundred experimentally screened combinations to tens of thousands of computationally tested combinations.The authors propose using these in silico screens to prioritize subsequent experimental testing, supported by withheld-experiment prediction accuracy.
  • Beyond the computational power of well-constrained and robustly derived network models: Model predictions can support investigation of drug-effect epistasis and resistance mechanisms, including combinations intended to counter resistance to targeted therapies.The paper presents this as a potential pre-clinical consequence rather than an established clinical outcome.

Methods and materials

The study perturbs BRAFV600E SKMEL-133 melanoma cells with targeted drugs and measures proteomic, phosphoproteomic, and phenotypic responses. These data are converted into executable signaling models using belief-propagation-guided inference, optimization, and steady-state simulation.

  • Choice of drugs and drug concentrations: Eight drugs targeting mainly MAPK or AKT/PI3K pathways were tested singly and in paired combinations in BRAFV600E SKMEL-133 cells.Drug concentrations were selected using Western-blot dose-response measurements of presumed targets or downstream effectors.
  • Setting up belief propagation calculations for network inference: The network models contain 25 nodes and use 44 experimental constraint sets derived from log-normalized proteomic and phenotypic responses.Each measured protein level was normalized against its corresponding no-drug control.
  • Inferring distinct, executable network models of signaling: Distinct network solutions are generated by BP-guided decimation, then edge parameters are optimized with Pineda/gradient descent before model ranking.The optimization relaxes discrete-edge and factorized-distribution assumptions while fitting both ε_i and α_i parameters.
  • Simulating signaling network models: Each inferred network is simulated under virtual perturbations until steady state using the DLSODE integration method.The simulations evaluate predicted responses after system variables stop changing within machine precision.
  • Activity nodes: Activity nodes represent perturbed molecular processes, such as kinase phosphorylation, when biochemical activities are not directly measured.Their baseline is set to zero relative to no-drug controls, and inhibitor effects are quantified through measured targets or downstream effectors.
  • Mathematical description of perturbations: The perturbation model treats a constant input as affecting both a node’s time derivative and its final steady-state value, while interaction parameters remain perturbation-independent.RNAi perturbations differ by influencing all nodes associated with the targeted gene, including total and phosphorylated products.
  • Toy Data: Toy networks with cascades, parallel chains, feed-forward motifs, and feedback motifs were generated to test BP against known interactions.Edge strengths were sampled between zero and a maximum strength of 2 after fixing network topology and interaction signs.

Supplementary material to: Perturbation biology- inferring signaling networks in cellular systems.

The supplementary analyses show that informative perturbation designs improve BP network inference, while discretization choices and comparisons with other methods impose important interpretive limits.

  • Perturbation strategies: Extended perturbation data yields higher recall faster and consistently higher precision than pair perturbations, indicating that more informative training patterns produce superior BP models.Pair perturbations are highly correlated, whereas the extended dataset produces less correlated patterns; adding pair experiments gives limited novel information while increasing inferred interactions.
  • Ideal-data benchmark: Ideal randomized data gives near-perfect BP inference, with zero false positives and only two false negatives against the generating network.The result demonstrates BP performance under maximally informative data, although such data are infeasible in biological systems.
  • Comparison with CoPIA: BP and CoPIA show partial agreement: 13 of 23 CoPIA interactions appear in at least two top BP models, while BP excludes only five interactions, including two high-probability CoPIA edges.The methods explore overlapping but non-identical regions of model space because of differences in search restrictions and topology modeling.
  • Comparison with CoPIA: BP assigns greater than 20% probability to 27 parameters, including eight interactions found among CoPIA’s top 100 models, while its top models achieve roughly 0.1 mean squared error.High-confidence incoming edges into MEK and pERK are prominent, but the authors do not establish why these nodes show this connectivity pattern.
  • Comparison with Gaussian Graphical Models: GGM comparisons show non-trivial overlap in inferred couplings with BP, but do not evaluate which method better reproduces trained responses or predicts responses to new perturbations.The GGM weight cutoff of 1 produces the highest-likelihood model for the analyzed data, while threshold selection determines network sparsity and interpretability.
  • Discretization sensitivity: Three-bin discretization produces near-zero entropy and all-or-nothing marginals, potentially excluding true non-zero values from sampled models.Entropy rises with bin count, but without accuracy or predictive-power comparisons the optimal discretization and benefits beyond 11 bins remain unclear.
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