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A Monte Carlo Cellular Simulation of the Repressilator
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A Monte Carlo Cellular Simulation of the Repressilator Oscillatory Network Seetal Erramilli 1,2 and Joel R. Stiles 1,3,4 1 Bioengineering and Bioinformatics Summer Institute, Department of Computational Biology, University of Pittsburgh 2 Department of Bioengineering, Pennsylvania State University 3 Mellon College of Science, Carnegie Mellon University 4 Senior Scientific Specialist, Computational Neuroscience, Biomedical Initiative, Pittsburgh Supercomputing Center Background Information: – The repressilator is a synthetic transcription regulator that consists of three proteins (LacI, tetR, cI) which mutually inhibit the production of one another, in a cyclic oscillatory manner. – Michael B. Elowitz, Stanislas Leibler ("A synthetic oscillatory network of transcriptional regulators", Nature, Vol 403, 20 January 2000) engineered the repressilator and had it grown in E. Coli. They compared actual experimental data with two types of models based on ordinary and stochastic differential equations. – Ordinary Differential Equations: Characterize the repressilator system to be in a properly mixed state at all times with relative diffusion constants of all molecules not taken into account. – Stochastic Differential Equations: Similar to ordinary however in this case via a Gillispie Monte Carlo algorithm it assumes a noise function. Fig 1: Diagram showing plasmid vector representation of the Repressilator Fig 2: Diagram showing schematic representation of the Repressilator gene network Comparison of MCell and Stochastic / Ordinary Differential Equations: The problem with using differential equations is that one assumes that the system is in a properly mixed state at all times. When looking at oscillatory networks, this assumption does not take into account the spatial evolution of the system since these types of networks have tendencies to locally evolve into inadequately mixed states. To overcome this problem the complex algorithms in MCell utilize Monte Carlo randomized movement principles characterized by specific normalized distributions, as well as two spatially realistic components of a reaction: ●The time it takes for molecules to move around and find each other in space ●Once molecules find one another the probability that their collision leads to a reaction Results and Data Analysis: The reaction rates of all interactions were rescaled to take into account MCell's inherent probability factors which assign higher probabilities for uni-molecular reactions compared to bimolecular reactions. Then the general conditions required for the system to exhibit unstability were implemented until oscillations were achieved. Fig 3: Graph showing protein and mRNA oscillations obtained from MCell Model Fig 4: Graph showing Protein oscillations from the stochastic differential equations model Comparisons between Monte Carlo and Differential Equations models: ●The magnitude of oscillations the protein species exhibit are very similar ●The period of oscillations are very different, due to the added spatial dimension in the MCell model, that looks at not only the rescaled reaction rates but the corresponding movements and velocities of all molecules. Biomedical Initiative Group Conclusions: As expected the repressilator exhibited oscillations of varying period but of similar magnititude. The time scales used in the differential equation model was in minutes whereas in the MCell model it was in microseconds. Although a global factor was applied to the reaction rates to increase the speed of the simulation, the period of oscillations cannot be reduced by that same factor to reproduce comparable oscillation periods between the two models. This is because even if the reaction rates are rescaled, one would also have to alter the relative diffusion rates of the particles to mimic the exact same conditions. The extent to which one must apply these scales is unknown, as the MCell simulation would have to run over a period of several hundred minutes which would require currently unachievable computational power. Acknowledgements: ● Bioengineering and Bioinformatics Summer Institute, Department of Computational Biology, University of Pittsburgh ● National Institutes of Health (NIH) and National Science Foundation (NSF) for funding of program ● Joel Stiles, Pittsburgh Supercomputing Center and Carnegie Mellon University ● Nicholas Morsillo, University of Pittsburgh ● Evan Kepner, University of Pittsburgh ● Duquense University To implement the repressilator system on a computationally achievable time scale, the reaction rates were scaled up selectively, taking into account the inherently higher probabilities of unimolecular versus bimolecular reactions. Fig 5: Graph showing oscillations from actual repressilator construction in E. Coli
Transcript of A Monte Carlo Cellular Simulation of the Repressilator
A Monte Carlo Cellular Simulation of the Repressilator Oscillatory Network Seetal Erramilli 1,2 and Joel R. Stiles 1,3,4 1 Bioengineering and Bioinformatics Summer Institute, Department of Computational Biology, University of Pittsburgh 2 Department of Bioengineering, Pennsylvania State University 3 Mellon College of Science, Carnegie Mellon University 4 Senior Scientific Specialist, Computational Neuroscience, Biomedical Initiative, Pittsburgh Supercomputing Center
Background Information: – The repressilator is a synthetic transcription regulator that consists of three proteins (LacI, tetR,
cI) which mutually inhibit the production of one another, in a cyclic oscillatory manner. – Michael B. Elowitz, Stanislas Leibler ("A synthetic oscillatory network of transcriptional
regulators", Nature, Vol 403, 20 January 2000) engineered the repressilator and had it grown in E. Coli. They compared actual experimental data with two types of models based on ordinary and stochastic differential equations. – Ordinary Differential Equations: Characterize the repressilator system to be in a properly mixed
state at all times with relative diffusion constants of all molecules not taken into account. – Stochastic Differential Equations: Similar to ordinary however in this case via a Gillispie Monte
Carlo algorithm it assumes a noise function.
Fig 1: Diagram showing plasmid vector representation of the Repressilator
Fig 2: Diagram showing schematic representation of the Repressilator gene network
Comparison of MCell and Stochastic / Ordinary Differential Equations: The problem with using differential equations is that one assumes that the system is in a properly
mixed state at all times. When looking at oscillatory networks, this assumption does not take into account the spatial evolution of the system since these types of networks have tendencies to locally evolve into inadequately mixed states.
To overcome this problem the complex algorithms in MCell utilize Monte Carlo randomized movement principles characterized by specific normalized distributions, as well as two spatially realistic components of a reaction:
●The time it takes for molecules to move around and find each other in space ●Once molecules find one another the probability that their collision leads to a reaction
Results and Data Analysis: The reaction rates of all interactions were rescaled to take into account MCell's inherent probability factors which assign higher probabilities for uni-molecular reactions compared to bimolecular reactions. Then the general conditions required for the system to exhibit unstability were implemented until oscillations were achieved.
Fig 3: Graph showing protein and mRNA oscillations obtained from MCell Model
Fig 4: Graph showing Protein oscillations from the stochastic differential equations model
Comparisons between Monte Carlo and Differential Equations models: ●The magnitude of oscillations the protein species exhibit are very similar ●The period of oscillations are very different, due to the added spatial dimension in the MCell model, that looks at not only the rescaled reaction rates but the corresponding movements and velocities of all molecules.
Biomedical Initiative Group
Conclusions: As expected the repressilator exhibited oscillations of varying period but of similar magnititude. The time scales used in the differential equation model was in minutes whereas in the MCell model it was in microseconds. Although a global factor was applied to the reaction rates to increase the speed of the simulation, the period of oscillations cannot be reduced by that same factor to reproduce comparable oscillation periods between the two models. This is because even if the reaction rates are rescaled, one would also have to alter the relative diffusion rates of the particles to mimic the exact same conditions. The extent to which one must apply these scales is unknown, as the MCell simulation would have to run over a period of several hundred minutes which would require currently unachievable computational power.
Acknowledgements: ● Bioengineering and Bioinformatics Summer Institute, Department of Computational Biology, University of Pittsburgh ● National Institutes of Health (NIH) and National Science Foundation (NSF) for funding of program ● Joel Stiles, Pittsburgh Supercomputing Center and Carnegie Mellon University ● Nicholas Morsillo, University of Pittsburgh ● Evan Kepner, University of Pittsburgh ● Duquense University
To implement the repressilator system on a computationally achievable time scale, the reaction rates were scaled up selectively, taking into account the inherently higher probabilities of unimolecular versus bimolecular reactions.
Fig 5: Graph showing oscillations from actual repressilator construction in E. Coli
