A perspective on bridging scales and design of models using low-dimensional manifolds and data-driven model inference

Systems in nature capable of collective behaviour are nonlinear, operating across several scales. Yet our ability to account for their collective dynamics differs in physics, chemistry and biology. Here, we briefly review the similarities and differences between mathematical modelling of adaptive living systems versus physico-chemical systems. We find that physics-based chemistry modelling and computational neuroscience have a shared interest in developing techniques for model reductions aiming at the identification of a reduced subsystem or slow manifold, capturing the effective dynamics. By contrast, as relations and kinetics between biological molecules are less characterized, current quantitative analysis under the umbrella of bioinformatics focuses on signal extraction, correlation, regression and machine-learning analysis. We argue that model reduction analysis and the ensuing identification of manifolds bridges physics and biology. Furthermore, modelling living systems presents deep challenges as how to reconcile rich molecular data with inherent modelling uncertainties (formalism, variables selection and model parameters). We anticipate a new generative data-driven modelling paradigm constrained by identified governing principles extracted from low-dimensional manifold analysis. The rise of a new generation of models will ultimately connect biology to quantitative mechanistic descriptions, thereby setting the stage for investigating the character of the model language and principles driving living systems. This article is part of the themed issue ‘Multiscale modelling at the physics–chemistry–biology interface’.

[1]  Marissa G. Saunders,et al.  Coarse-graining methods for computational biology. , 2013, Annual review of biophysics.

[2]  James G. King,et al.  Reconstruction and Simulation of Neocortical Microcircuitry , 2015, Cell.

[3]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[4]  Karl J. Friston,et al.  A Free Energy Principle for Biological Systems. , 2012, Entropy.

[5]  Jesper Tegnér,et al.  Mechanism for top-down control of working memory capacity , 2009, Proceedings of the National Academy of Sciences.

[6]  Ruth Nussinov,et al.  A second molecular biology revolution? The energy landscapes of biomolecular function. , 2014, Physical chemistry chemical physics : PCCP.

[7]  A. Barabasi,et al.  Network biology: understanding the cell's functional organization , 2004, Nature Reviews Genetics.

[8]  O. Wolkenhauer Why model? , 2013, Front. Physiol..

[9]  J. Hertz,et al.  Mean field theory for nonequilibrium network reconstruction. , 2010, Physical review letters.

[10]  Gilles Clermont,et al.  Computational disease modeling – fact or fiction? , 2009, BMC Systems Biology.

[11]  Karl J. Friston The free-energy principle: a unified brain theory? , 2010, Nature Reviews Neuroscience.

[12]  A. M. Turing,et al.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[13]  Jesper Tegnér,et al.  Decoding complex biological networks - tracing essential and modulatory parameters in complex and simplified models of the cell cycle , 2011, BMC Systems Biology.

[14]  The Unreasonable Effectiveness of Mat hematics in the Natural Sciences , 2006 .

[15]  J. Tegnér,et al.  Perturbations to uncover gene networks. , 2007, Trends in genetics : TIG.

[16]  Manfred Dietrich Laubichler,et al.  Conrad H. Waddington: Towards a Theoretical Biology , 2008 .

[17]  J. Rinzel Excitation dynamics: insights from simplified membrane models. , 1985, Federation proceedings.

[18]  Martin Karplus,et al.  Development of Multiscale Models for Complex Chemical Systems: From H + H2 to Biomolecules (Nobel Lecture) , 2014 .

[19]  Wulfram Gerstner,et al.  Neuronal Dynamics: From Single Neurons To Networks And Models Of Cognition , 2014 .

[20]  Hector Zenil,et al.  Methods of information theory and algorithmic complexity for network biology. , 2014, Seminars in cell & developmental biology.

[21]  Olaf Wolkenhauer,et al.  The search for organizing principles as a cure against reductionism in systems medicine , 2013, The FEBS journal.

[22]  U. Alon Network motifs: theory and experimental approaches , 2007, Nature Reviews Genetics.

[23]  Michael J. Berry,et al.  Searching for Collective Behavior in a Large Network of Sensory Neurons , 2013, PLoS Comput. Biol..

[24]  Julio R. Banga,et al.  Reverse engineering and identification in systems biology: strategies, perspectives and challenges , 2014, Journal of The Royal Society Interface.

[25]  W. Bialek Biophysics: Searching for Principles , 2012 .

[26]  James P. Crutchfield,et al.  The Elusive Present: Hidden Past and Future Dependency and Why We Build Models , 2015, Physical review. E.

[27]  E. Aurell,et al.  Dynamics inside the cancer cell attractor reveal cell heterogeneity, limits of stability, and escape , 2016, Proceedings of the National Academy of Sciences.

[28]  Shunkai Fu,et al.  Markov Blanket based Feature Selection: A Review of Past Decade , 2010 .

[29]  Guillaume Crevecoeur,et al.  Sparse identification of nonlinear dynamical systems from data , 2018 .

[30]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[31]  B. Rannala,et al.  Molecular phylogenetics: principles and practice , 2012, Nature Reviews Genetics.

[32]  Hector Zenil,et al.  Information Theory and Computational Thermodynamics: Lessons for Biology from Physics , 2012, Inf..

[33]  Jesper Tegnér,et al.  Consistent Feature Selection for Pattern Recognition in Polynomial Time , 2007, J. Mach. Learn. Res..

[34]  S. Brunton,et al.  Discovering governing equations from data by sparse identification of nonlinear dynamical systems , 2015, Proceedings of the National Academy of Sciences.

[35]  Jeremy L. England,et al.  Statistical physics of self-replication. , 2012, The Journal of chemical physics.

[36]  J. Pearl Causal Thinking in the Twilight Zone , 2021 .

[37]  Michael Levitt,et al.  Training-free atomistic prediction of nucleosome occupancy , 2014, Proceedings of the National Academy of Sciences.

[38]  E. Wigner The Unreasonable Effectiveness of Mathematics in the Natural Sciences (reprint) , 1960 .

[39]  L. Abbott,et al.  Theoretical Neuroscience Rising , 2008, Neuron.

[40]  M. Karplus Development of multiscale models for complex chemical systems: from H+H₂ to biomolecules (Nobel Lecture). , 2014, Angewandte Chemie.

[41]  Hector Zenil,et al.  Evaluating Network Inference Methods in Terms of Their Ability to Preserve the Topology and Complexity of Genetic Networks , 2015, Seminars in cell & developmental biology.

[42]  James P. Crutchfield,et al.  Computational Mechanics of Input–Output Processes: Structured Transformations and the ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \ , 2014, Journal of Statistical Physics.

[43]  Rui-Sheng Wang,et al.  Boolean modeling in systems biology: an overview of methodology and applications , 2012, Physical biology.

[44]  James P. Crutchfield,et al.  Computational Mechanics of Input-Output Processes: Structured transformations and the ε-transducer , 2014, ArXiv.

[45]  R. FitzHugh Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[46]  Yuji Sugita,et al.  Complete atomistic model of a bacterial cytoplasm for integrating physics, biochemistry, and systems biology. , 2015, Journal of molecular graphics & modelling.

[47]  Hector Zenil,et al.  Quantifying loss of information in network-based dimensionality reduction techniques , 2015, J. Complex Networks.

[48]  Hector Zenil,et al.  The Information-theoretic and Algorithmic Approach to Human, Animal and Artificial Cognition , 2015, ArXiv.

[49]  A. Turing The chemical basis of morphogenesis , 1990 .

[50]  William Bialek,et al.  Information processing in living systems , 2014, 1412.8752.

[51]  Jonathan R. Karr,et al.  A Whole-Cell Computational Model Predicts Phenotype from Genotype , 2012, Cell.

[52]  Evan O. Paull,et al.  Inferring causal molecular networks: empirical assessment through a community-based effort , 2016, Nature Methods.

[53]  Albert Compte,et al.  Workflow for generating competing hypothesis from models with parameter uncertainty , 2011, Interface Focus.

[54]  J. Tyson,et al.  Design principles of biochemical oscillators , 2008, Nature Reviews Molecular Cell Biology.

[55]  Huxley Af,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve. 1952. , 1990 .

[56]  J. Tegnér,et al.  Integrated approaches to uncovering transcription regulatory networks in mammalian cells. , 2008, Genomics.

[57]  Amy K. Schmid,et al.  A Predictive Model for Transcriptional Control of Physiology in a Free Living Cell , 2007, Cell.

[58]  Liesbet Geris,et al.  Uncertainty in Biology , 2016 .

[59]  Diogo M. Camacho,et al.  Wisdom of crowds for robust gene network inference , 2012, Nature Methods.

[60]  Y. Kosmann-Schwarzbach The Noether Theorems , 2011 .

[61]  T. van Mourik,et al.  Density functional theory across chemistry, physics and biology , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[62]  Yangyang Zhao,et al.  BioModels: ten-year anniversary , 2014, Nucleic Acids Res..

[63]  J. Ludden,et al.  Principles and Practice , 1998, Community-based Learning and Social Movements.

[64]  Alfio Quarteroni,et al.  A vision and strategy for the virtual physiological human: 2012 update , 2013, Interface Focus.

[65]  Kenneth Showalter,et al.  From chemical systems to systems chemistry: Patterns in space and time. , 2015, Chaos.

[66]  Ioannis Tsamardinos,et al.  Probabilistic Computational Causal Discovery for Systems Biology , 2016 .

[67]  Hod Lipson,et al.  Distilling Free-Form Natural Laws from Experimental Data , 2009, Science.

[68]  David Gomez-Cabrero,et al.  Data integration in the era of omics: current and future challenges , 2014, BMC Systems Biology.

[69]  Emmanuel Candes,et al.  Stable signal recovery from incomplete observations , 2005, SPIE Optics + Photonics.

[70]  George M. Bodner,et al.  MODELS AND MODELING , 2005 .

[71]  K. Dill,et al.  Principles of maximum entropy and maximum caliber in statistical physics , 2013 .