Maximum Entropy Framework for Predictive Inference of Cell Population Heterogeneity and Responses in Signaling Networks.

Predictive models of signaling networks are essential for understanding cell population heterogeneity and designing rational interventions in disease. However, using computational models to predict heterogeneity of signaling dynamics is often challenging because of the extensive variability of biochemical parameters across cell populations. Here, we describe a maximum entropy-based framework for inference of heterogeneity in dynamics of signaling networks (MERIDIAN). MERIDIAN estimates the joint probability distribution over signaling network parameters that is consistent with experimentally measured cell-to-cell variability of biochemical species. We apply the developed approach to investigate the response heterogeneity in the EGFR/Akt signaling network. Our analysis demonstrates that a significant fraction of cells exhibits high phosphorylated Akt (pAkt) levels hours after EGF stimulation. Our findings also suggest that cells with high EGFR levels predominantly contribute to the subpopulation of cells with high pAkt activity. We also discuss how MERIDIAN can be extended to accommodate various experimental measurements.

[1]  Kingshuk Ghosh,et al.  Perspective: Maximum caliber is a general variational principle for dynamical systems. , 2017, The Journal of chemical physics.

[2]  David Stewart,et al.  Connecting the dots across time: reconstruction of single-cell signalling trajectories using time-stamped data , 2016, Royal Society Open Science.

[3]  Jayajit Das,et al.  Maximum Entropy Estimation of Probability Distribution of Variables in Higher Dimensions from Lower Dimensional Data , 2015, Entropy.

[4]  D. Gross,et al.  Heterogeneity of epidermal growth factor binding kinetics on individual cells. , 1997, Biophysical journal.

[5]  C. Sawyers,et al.  The phosphatidylinositol 3-Kinase–AKT pathway in human cancer , 2002, Nature Reviews Cancer.

[6]  Ken A Dill,et al.  Communication: Maximum caliber is a general variational principle for nonequilibrium statistical mechanics. , 2015, The Journal of chemical physics.

[7]  Michael J. Berry,et al.  Ising models for networks of real neurons , 2006, q-bio/0611072.

[8]  Dennis Vitkup,et al.  Diverse types of genetic variation converge on functional gene networks involved in schizophrenia , 2012, Nature Neuroscience.

[9]  T. Tiganis.  Protein Tyrosine Phosphatases: Dephosphorylating the Epidermal Growth Factor Receptor , 2002, IUBMB life.

[10]  M. Stein.  Large sample properties of simulations using latin hypercube sampling , 1987 .

[11]  Frank Allgöwer,et al.  Identification of models of heterogeneous cell populations from population snapshot data , 2011, BMC Bioinformatics.

[12]  S Cocco,et al.  Large pseudocounts and L2-norm penalties are necessary for the mean-field inference of Ising and Potts models. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  A. Oudenaarden,et al.  Nature, Nurture, or Chance: Stochastic Gene Expression and Its Consequences , 2008, Cell.

[14]  J. Timmer,et al.  Systems biology: experimental design , 2009, The FEBS journal.

[15]  Jeremy L. Muhlich,et al.  Properties of cell death models calibrated and compared using Bayesian approaches , 2013, Molecular systems biology.

[16]  Roy S Herbst,et al.  Review of epidermal growth factor receptor biology. , 2004, International journal of radiation oncology, biology, physics.

[17]  Michael J. Berry,et al.  Weak pairwise correlations imply strongly correlated network states in a neural population , 2005, Nature.

[18]  C. Futter,et al.  EGF receptor trafficking: consequences for signaling and cancer , 2014, Trends in cell biology.

[19]  Zackary R. Kenz,et al.  A review of selected techniques in inverse problem nonparametric probability distribution estimation , 2012 .

[20]  Peter K. Sorger,et al.  Conservation of protein abundance patterns reveals the regulatory architecture of the EGFR-MAPK pathway , 2016, Science Signaling.

[21]  T. Hwa,et al.  Identification of direct residue contacts in protein–protein interaction by message passing , 2009, Proceedings of the National Academy of Sciences.

[22]  Peter K. Sorger,et al.  Receptor-Based Mechanism of Cell Memory and Relative Sensing in Mammalian Signaling Networks , 2017, bioRxiv.

[23]  A. Saliba,et al.  Single-cell RNA-seq: advances and future challenges , 2014, Nucleic acids research.

[24]  Purushottam D Dixit,et al.  Quantifying extrinsic noise in gene expression using the maximum entropy framework. , 2013, Biophysical journal.

[25]  A. Singh,et al.  Single-cell protein analysis. , 2012, Current opinion in biotechnology.

[26]  Mario Roederer,et al.  Single-cell technologies for monitoring immune systems , 2014, Nature Immunology.

[27]  Jan Hasenauer,et al.  Mathematical modeling of variability in intracellular signaling , 2019, Current Opinion in Systems Biology.

[28]  Roland Eils,et al.  Correlated receptor transport processes buffer single-cell heterogeneity , 2017, PLoS Comput. Biol..

[29]  Simona Cocco,et al.  Inverse statistical physics of protein sequences: a key issues review , 2017, Reports on progress in physics. Physical Society.

[30]  Sang-Cheol Seok,et al.  Cell responses only partially shape cell-to-cell variations in protein abundances in Escherichia coli chemotaxis , 2013, Proceedings of the National Academy of Sciences.

[31]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[32]  Guo-Cheng Yuan,et al.  Broadly heterogeneous activation of the master regulator for sporulation in Bacillus subtilis , 2010, Proceedings of the National Academy of Sciences.

[33]  Sabrina L Spencer,et al.  Non-genetic Cell-to-cell Variability and the Consequences for Pharmacology This Review Comes from a Themed Issue on Omics Edited the Distribution of Protein Abundance and Resulting Variability in Phenotype Measuring Cell-to-cell Variation , 2022 .

[34]  G. Nolan,et al.  In silico modeling identifies CD45 as a regulator of IL-2 synergy in the NKG2D-mediated activation of immature human NK cells , 2017, Science Signaling.

[35]  Daniel T Gillespie,et al.  Stochastic simulation of chemical kinetics. , 2007, Annual review of physical chemistry.

[36]  J. Lygeros,et al.  Moment-based inference predicts bimodality in transient gene expression , 2012, Proceedings of the National Academy of Sciences.

[37]  K. Dill,et al.  Inferring Transition Rates of Networks from Populations in Continuous-Time Markov Processes. , 2015, Journal of chemical theory and computation.

[38]  Eugenio Cinquemani,et al.  What Population Reveals about Individual Cell Identity: Single-Cell Parameter Estimation of Models of Gene Expression in Yeast , 2016, PLoS Comput. Biol..

[39]  Purushottam D. Dixit,et al.  Communication: Introducing prescribed biases in out-of-equilibrium Markov models , 2018 .

[40]  John P. Cunningham,et al.  Maximum Entropy Flow Networks , 2017, ICLR.

[41]  R. Preuss,et al.  Maximum entropy and Bayesian data analysis: Entropic prior distributions. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[42]  K. M. Nicholson,et al.  The protein kinase B/Akt signalling pathway in human malignancy. , 2002, Cellular signalling.

[43]  C. Gillespie.  Moment-closure approximations for mass-action models. , 2009, IET systems biology.

[44]  David H Perlman,et al.  Single-cell mass-spectrometry quantifies the emergence of macrophage heterogeneity , 2019 .

[45]  Peter G Wolynes,et al.  Transferable model for chromosome architecture , 2016, Proceedings of the National Academy of Sciences.

[46]  Jan Hasenauer,et al.  Estimation of biochemical network parameter distributions in cell populations , 2009, 0905.1191.

[47]  D. Lauffenburger,et al.  Input–output behavior of ErbB signaling pathways as revealed by a mass action model trained against dynamic data , 2009, Molecular systems biology.

[48]  D. Pe’er,et al.  Trajectories of cell-cycle progression from fixed cell populations , 2015, Nature Methods.

[49]  B. Berne,et al.  Spectral gap optimization of order parameters for sampling complex molecular systems , 2015, Proceedings of the National Academy of Sciences.

[50]  W. Bialek,et al.  Maximum entropy models for antibody diversity , 2009, Proceedings of the National Academy of Sciences.

[51]  Roland Eils,et al.  Optimal Experimental Design for Parameter Estimation of a Cell Signaling Model , 2009, PLoS Comput. Biol..

[52]  N. Slavov,et al.  SCoPE-MS: mass spectrometry of single mammalian cells quantifies proteome heterogeneity during cell differentiation , 2017, Genome Biology.

[53]  D. Lauffenburger,et al.  Quantitative analysis of pathways controlling extrinsic apoptosis in single cells. , 2008, Molecular cell.

[54]  P. Sorger,et al.  Non-genetic origins of cell-to-cell variability in TRAIL-induced apoptosis , 2009, Nature.

[55]  Hao Wu,et al.  Combining experimental and simulation data of molecular processes via augmented Markov models , 2017, Proceedings of the National Academy of Sciences.

[56]  Rodney W. Johnson,et al.  Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy , 1980, IEEE Trans. Inf. Theory.

[57]  Fabian J. Theis,et al.  ODE Constrained Mixture Modelling: A Method for Unraveling Subpopulation Structures and Dynamics , 2014, PLoS Comput. Biol..

[58]  Kresten Lindorff-Larsen,et al.  Conformational ensembles of RNA oligonucleotides from integrating NMR and molecular simulations , 2018, Science Advances.

[59]  Peter K. Sorger,et al.  Sporadic ERK pulses drive non-genetic resistance in drug-adapted BRAFV600E melanoma cells , 2019, bioRxiv.

[60]  Jan Hasenauer,et al.  A Hierarchical, Data-Driven Approach to Modeling Single-Cell Populations Predicts Latent Causes of Cell-To-Cell Variability. , 2018, Cell systems.

[61]  Jennifer C. Waters,et al.  Accuracy and precision in quantitative fluorescence microscopy , 2009, The Journal of cell biology.

[62]  E. Hirsch,et al.  PI3K/AKT signaling pathway and cancer: an updated review , 2014, Annals of medicine.

[63]  Eli R. Zunder,et al.  Highly multiplexed simultaneous detection of RNAs and proteins in single cells , 2016, Nature Methods.

[64]  A. Toker,et al.  AKT/PKB Signaling: Navigating the Network , 2017, Cell.

[65]  Johan Karlsson,et al.  Heterogeneous kinetics of AKT signaling in individual cells are accounted for by variable protein concentration , 2012, Front. Physio..

[66]  Sean R. Bittner,et al.  Approximating exponential family models (not single distributions) with a two-network architecture , 2019, ArXiv.

[67]  Steffen Waldherr,et al.  Estimation methods for heterogeneous cell population models in systems biology , 2018, Journal of The Royal Society Interface.

[68]  R. Grima,et al.  Linear mapping approximation of gene regulatory networks with stochastic dynamics , 2018, Nature Communications.