The Utilization of Monte Carlo Techniques to Simulate the Repressilator: A Cyclic Oscillatory System Seetal Erramilli 1,2 and Joel R. Stiles 1,3,4 1 Bioengineering and Bioinformatics Summer Institute, Department of Computational Biology, University of Pittsburgh, Pittsburgh, PA 15261 2 Department of Bioengineering, Pennsylvania State University, University Park, PA 16802 3 Mellon College of Science, Carnegie Mellon University, Pittsburgh, PA 15213 4 Biomedical Initiative, Pittsburgh Supercomputing Center, Pittsburgh, PA 15213 Engineering of Repressilator Elowitz and Leibler, A synthetic oscillatory network of transcriptional regulators, Nature, Vol 403, Jan. 2000. The repressilator is a synthetic transcriptional regulator consisting of 3 proteins (LacI, TetR, ci) that mutually inhibit one another in a cyclic oscillatory manner. The repressilator was designed in a plasmid vector form, introduced into E. coli, and by tracking fluorescence the oscillations of the protein concentrations were determined experimentally in single E. coli. Figure from Nature article showing repressilator oscillations Schematic representation of the repressilator gene network 1
Modeling of Repressilator Elowitz and Leibler modeled the repressilator network using deterministic and stochastic methods Deterministic (Differential Equations): α no. of protein copies per cell produced from a given promoter in absence of saturating amounts of repressor β ratio of protein to mrna half life Treats the concentrations of protein and mrna as continuous dynamical variables. Stochastic (Gillespie Monte Carlo): Simulates stochastic chemical transitions of discrete molecules, but does not include locations and movements of the molecules (assumes wellmixed conditions) Results of Deterministic and Stochastic Approaches Deterministic, Continuous Method Stochastic, Discrete Method Both methods assume that the system is well mixed. However, networks with feedback loops can evolve locally, so a more realistic simulation must include the spatial evolution of the system. 2
MCell Implementation MCell is a Monte Carlo Cellular Simulation program MCell incorporates two major components of molecular interactions: Diffusion of molecules in arbitrary spaces Reactions between diffusing molecules subsequent to collisions in space A realistic 3-D random walk algorithm is used to simulate molecular diffusion The simulation depicts molecule positions, movements and interactions with spatial realism Results of MCell Simulation Reaction rates of all interactions were increased by different scaling factors (due to probability differences between unimolecular and bimolecular reactions) This was done until oscillations were achieved Figure showing MCell s Protein and mrna oscillations 3
Comparing MCell Simulation to E. coli Experiments Protein Oscillations in E. coli between different cells under different conditions Protein Oscillations in MCell 1. Oscillations exhibit varying magnitude and period in both cases 2. The time scales are dramatically different (minutes vs. μs) Comparing MCell Simulation to E. coli Experiments Protein Oscillations in single E. coli Protein Oscillations in MCell 1. Oscillations exhibit varying magnitude and period in both cases 2. The time scales are dramatically different (minutes vs. μs) 4
Conclusions The varying magnitudes and periods of oscillations exhibited in MCell qualitatively match the oscillations exhibited in E. coli Spatial realism is important since the intracellular structure of E. coli is intricately complex The MCell simulated time differs to the actual oscillation period of the network by several orders of magnitude Potential Extensions Develop the MCell simulation for a much longer time scale to more accurately represent real network Further improve spatial dynamics by incorporating actual protein, mrna and DNA structures as well as detailed E. coli surfaces Current obstacle preventing such implementations is insufficient computational power Movie showing Protein oscillations exhibited in MCell Simulation 5
Acknowledgements I d like to thank the following individuals and organizations: National Institutes of Health (NIH) and National Science Foundation (NSF) Rajan Munshi, Coordinator of Bioengineering and Bioinformatics Summer Institute (BBSI) Joel Stiles, Pittsburgh Supercomputing Center & Carnegie Mellon University John Pattillo, Pittsburgh Supercomputing Center Nicholas Morsillo, University of Pittsburgh Evan Kepner, University of Pittsburgh All participants of BBSI References Michael B. Elowitz, Stanislas Leibler, "A synthetic oscillatory network of transcriptional regulators", NATURE, VOL 403, 20 January 2000 Nuri A. Temiz, Lee Wei Yang, "Repressilator: A negative feedback system. Deterministic and stochastic simulations Tom Chou, "Ribosome recycling, diffusion, and mrna loop formation in translational regulation", Biophysical Journal, Volume 85, August 2003 Jordi Garcia-Ojalvo, Michael B. Elowitz, Steven H. Strogatz, "Modeling a synthetic multicellular clock: Repressilators coupled by quorum sensing", PNAS, July 27, 2004, VOL 101 Tianshou Zhou, Luonan Chen, Ruiqi Wang, Kazuyuki Aihara, Intercellular Communications Induced by Random Fluctuations, Genome Informatics, 2004, 15(2) Jeff Hasty, David Mcmillen, J. J. Collins, Engineered gene circuits, NATURE, VOL 420, 14 November 2002 6