Contraction Theory for Systems Biology

In this chapter, we present a theoretical framework for the analysis, synchronization and control of biochemical circuits and systems modeled by means of ODEs. The methodology is based on the use of contraction theory, a powerful concept from the theory of dynamical systems, ensuring convergence of all trajectories of a system of interest towards each other. After introducing contraction theory, we present some application to biochemical networks. Specifically, we introduce a graphical approach to verify if a system is contracting and apply it to synthesize networks of self-synchronizing Repressilators. We then present a more general analysis of quorum sensing networks.

[1]  Jean-Jacques E. Slotine,et al.  On partial contraction analysis for coupled nonlinear oscillators , 2004, Biological Cybernetics.

[2]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .

[3]  D. C. Lewis.  Metric Properties of Differential Equations , 1949 .

[4]  Bonnie L Bassler,et al.  Observing bacteria through the lens of social evolution , 2008, Journal of biology.

[5]  Dileep George,et al.  Towards a Mathematical Theory of Cortical Micro-circuits , 2009, PLoS Comput. Biol..

[6]  Mario di Bernardo,et al.  How to Synchronize Biological Clocks , 2009, J. Comput. Biol..

[7]  Shimon Ullman,et al.  Combined Top-Down/Bottom-Up Segmentation , 2008, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[8]  M. Elowitz,et al.  Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Jean-Jacques E. Slotine,et al.  Nonlinear process control using contraction theory , 2000 .

[10]  Nathan van de Wouw,et al.  Convergent dynamics, a tribute to Boris Pavlovich Demidovich , 2004, Syst. Control. Lett..

[11]  B. Bassler,et al.  Quorum sensing in bacteria. , 2001, Annual review of microbiology.

[12]  Eduardo Sontag,et al.  Modular cell biology: retroactivity and insulation , 2008, Molecular systems biology.

[13]  Joao B Xavier,et al.  The Evolution of Quorum Sensing in Bacterial Biofilms , 2008, PLoS biology.

[14]  Mario di Bernardo,et al.  Contraction Theory and Master Stability Function: Linking Two Approaches to Study Synchronization of Complex Networks , 2009, IEEE Transactions on Circuits and Systems II: Express Briefs.

[15]  Y Sakaki,et al.  Entrainment of the circadian clock in the liver by feeding. , 2001, Science.

[16]  Rita Toth,et al.  Collective behavior of a population of chemically coupled oscillators. , 2006, The journal of physical chemistry. B.

[17]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[18]  J. Collins,et al.  Programmable cells: interfacing natural and engineered gene networks. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[19]  Célia Baroux,et al.  Positive darwinian selection at the imprinted MEDEA locus in plants , 2007, Nature.

[20]  Nancy Kopell,et al.  Synchrony in a Population of Hysteresis-Based Genetic Oscillators , 2004, SIAM J. Appl. Math..

[21]  A. Politi,et al.  Collective atomic recoil laser as a synchronization transition. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  W. Lohmiller,et al.  Contraction analysis of non-linear distributed systems , 2005 .

[23]  B. Bassler,et al.  Bacterial quorum-sensing network architectures. , 2009, Annual review of genetics.

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

[25]  P. Hartman.  On Stability in the Large for Systems of Ordinary Differential Equations , 1961, Canadian Journal of Mathematics.

[26]  Jean-Jacques E. Slotine,et al.  Visual Grouping by Neural Oscillator Networks , 2009, IEEE Transactions on Neural Networks.

[27]  David Angeli,et al.  A Lyapunov approach to incremental stability properties , 2002, IEEE Trans. Autom. Control..

[28]  J. Collins,et al.  Construction of a genetic toggle switch in Escherichia coli , 2000, Nature.

[29]  Jochen Braun,et al.  Bistable Perception Modeled as Competing Stochastic Integrations at Two Levels , 2009, PLoS Comput. Biol..

[30]  Kirsten Jung,et al.  Heterogeneity in quorum sensing‐regulated bioluminescence of Vibrio harveyi , 2009, Molecular microbiology.

[31]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[32]  Mario di Bernardo,et al.  Global Entrainment of Transcriptional Systems to Periodic Inputs , 2009, PLoS Comput. Biol..

[33]  R. Weiss,et al.  Programmed population control by cell–cell communication and regulated killing , 2004, Nature.

[34]  Nicolas Tabareau,et al.  How Synchronization Protects from Noise , 2007, 0801.0011.

[35]  M. Alexander,et al.  Principles of Neural Science , 1981 .

[36]  Sean M Montgomery,et al.  The Effect of Spatially Inhomogeneous Extracellular Electric Fields on Neurons , 2010, The Journal of Neuroscience.

[37]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[38]  Bijan Pesaran,et al.  Temporal structure in neuronal activity during working memory in macaque parietal cortex , 2000, Nature Neuroscience.

[39]  Jean-Jacques E. Slotine,et al.  Modular stability tools for distributed computation and control , 2003 .

[40]  Mario di Bernardo,et al.  A Graphical Approach to Prove Contraction of Nonlinear Circuits and Systems , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.

[41]  D. Fidock,et al.  Arresting malaria parasite egress from infected red blood cells. , 2008, Nature chemical biology.

[42]  Giovanni Russo,et al.  Global convergence of quorum-sensing networks. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  J. Jouffroy.  Some ancestors of contraction analysis , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[44]  M. Elowitz,et al.  A synthetic oscillatory network of transcriptional regulators , 2000, Nature.

[45]  Pierre Yger,et al.  Network-State Modulation of Power-Law Frequency-Scaling in Visual Cortical Neurons , 2009, PLoS Comput. Biol..

[46]  Jeff Hasty,et al.  Stochastic Emergence of Groupthink , 2010, Science.

[47]  Thomas Gregor,et al.  The Onset of Collective Behavior in Social Amoebae , 2010, Science.

[48]  Nicolas Tabareau,et al.  A Contraction Theory Approach to Stochastic Incremental Stability , 2007, IEEE Transactions on Automatic Control.

[49]  D. Beebe,et al.  Physics and applications of microfluidics in biology. , 2002, Annual review of biomedical engineering.

[50]  Mario di Bernardo,et al.  An algorithm for the construction of synthetic self synchronizing biological circuits , 2009, 2009 IEEE International Symposium on Circuits and Systems.