Fast stochastic simulation of biochemical reaction systems by\ud alternative formulations of the Chemical Langevin Equation

The Chemical Langevin Equation (CLE), which is a stochastic differential equation (SDE) driven by a multidimensional Wiener process, acts as a bridge between the discrete Stochastic Simulation Algorithm and the deterministic reaction rate equation when simulating (bio)chemical kinetics. The CLE model is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. The contribution of this work is that we observe and explore that the CLE is not a single equation, but a parametric family of equations, all of which give the same finite-dimensional distribution of the variables. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation. On the practical side, we show that in the case where there are m1 pairs of reversible reactions and m2 irreversible reactions only m1+m2 Wiener processes are required in the formulation of the CLE, whereas the standard approach uses 2m1 + m2. We illustrate our findings by considering alternative formulations of the CLE for a\ud HERG ion channel model and the Goldbeter–Koshland switch. We show that there are considerable computational savings when using our insights

Linear mapping approximation of gene regulatory networks with stochastic dynamics

The intractability of most stochastic models of gene regulatory networks (GRNs) limits their utility. Here, the authors present a linear-mapping approximation mapping models onto simpler ones, giving approximate but accurate analytic or semi- analytic solutions for a wide range of model GRNs

Dynamics of Genome Rearrangement in Bacterial Populations

Genome structure variation has profound impacts on phenotype in organisms ranging from microbes to humans, yet little is known about how natural selection acts on genome arrangement. Pathogenic bacteria such as Yersinia pestis, which causes bubonic and pneumonic plague, often exhibit a high degree of genomic rearrangement. The recent availability of several Yersinia genomes offers an unprecedented opportunity to study the evolution of genome structure and arrangement. We introduce a set of statistical methods to study patterns of rearrangement in circular chromosomes and apply them to the Yersinia. We constructed a multiple alignment of eight Yersinia genomes using Mauve software to identify 78 conserved segments that are internally free from genome rearrangement. Based on the alignment, we applied Bayesian statistical methods to infer the phylogenetic inversion history of Yersinia. The sampling of genome arrangement reconstructions contains seven parsimonious tree topologies, each having different histories of 79 inversions. Topologies with a greater number of inversions also exist, but were sampled less frequently. The inversion phylogenies agree with results suggested by SNP patterns. We then analyzed reconstructed inversion histories to identify patterns of rearrangement. We confirm an over-representation of “symmetric inversions”—inversions with endpoints that are equally distant from the origin of chromosomal replication. Ancestral genome arrangements demonstrate moderate preference for replichore balance in Yersinia. We found that all inversions are shorter than expected under a neutral model, whereas inversions acting within a single replichore are much shorter than expected. We also found evidence for a canonical configuration of the origin and terminus of replication. Finally, breakpoint reuse analysis reveals that inversions with endpoints proximal to the origin of DNA replication are nearly three times more frequent. Our findings represent the first characterization of genome arrangement evolution in a bacterial population evolving outside laboratory conditions. Insight into the process of genomic rearrangement may further the understanding of pathogen population dynamics and selection on the architecture of circular bacterial chromosomes

Multi-objective optimization framework to obtain model-based guidelines for tuning biological synthetic devices: an adaptive network case

Background: Model based design plays a fundamental role in synthetic biology. Exploiting modularity, i.e. using biological parts and interconnecting them to build new and more complex biological circuits is one of the key issues. In this context, mathematical models have been used to generate predictions of the behavior of the designed device. Designers not only want the ability to predict the circuit behavior once all its components have been determined, but also to help on the design and selection of its biological parts, i.e. to provide guidelines for the experimental implementation. This is tantamount to obtaining proper values of the model parameters, for the circuit behavior results from the interplay between model structure and parameters tuning. However, determining crisp values for parameters of the involved parts is not a realistic approach. Uncertainty is ubiquitous to biology, and the characterization of biological parts is not exempt from it. Moreover, the desired dynamical behavior for the designed circuit usually results from a trade-off among several goals to be optimized. Results: We propose the use of a multi-objective optimization tuning framework to get a model-based set of guidelines for the selection of the kinetic parameters required to build a biological device with desired behavior. The design criteria are encoded in the formulation of the objectives and optimization problem itself. As a result, on the one hand the designer obtains qualitative regions/intervals of values of the circuit parameters giving rise to the predefined circuit behavior; on the other hand, he obtains useful information for its guidance in the implementation process. These parameters are chosen so that they can effectively be tuned at the wet-lab, i.e. they are effective biological tuning knobs. To show the proposed approach, the methodology is applied to the design of a well known biological circuit: a genetic incoherent feed-forward circuit showing adaptive behavior. Conclusion: The proposed multi-objective optimization design framework is able to provide effective guidelines to tune biological parameters so as to achieve a desired circuit behavior. Moreover, it is easy to analyze the impact of the context on the synthetic device to be designed. That is, one can analyze how the presence of a downstream load influences the performance of the designed circuit, and take it into account.Research in this area is partially supported by Spanish government and European Union (FEDER-CICYT DPI2011-28112-C04-01, and DPI2014-55276-C5-1-R). Yadira Boada thanks grant FPI/2013-3242 of Universitat Politecnica de Valencia; Gilberto Reynoso-Meza gratefully acknowledges the partial support provided by the postdoctoral fellowship BJT-304804/2014-2 from the National Council of Scientific and Technologic Development of Brazil (CNPq) for the development of this work. We are grateful to Alejandra Gonzalez-Bosca for her collaboration on this topic while doing her Bachelor thesis, and to Dr. Jose Luis Pitarch from Universidad de Valladolid for his advise in algorithmic implementations and for proof reading the manuscript.Boada Acosta, YF.; Reynoso Meza, G.; Picó Marco, JA.; Vignoni, A. (2016). Multi-objective optimization framework to obtain model-based guidelines for tuning biological synthetic devices: an adaptive network case. BMC Systems Biology. 10:1-19. https://doi.org/10.1186/s12918-016-0269-0S11910ERASynBio. Next steps for european synthetic biology: a strategic vision from erasynbio. Report, ERASynBio. 2014. https://www.erasynbio.eu/lw_resource/datapool/_items/item_58/erasynbiostrategicvision.pdf .Way J, Collins J, Keasling J, Silver P. Integrating biological redesign: Where synthetic biology came from and where it needs to go. Cell. 2014; 157(1):151–61.Canton B, Labno A, Endy D. Refinement and standardization of synthetic biological parts and devices. Nat Biotechnol. 2008; 26(7):787–93.De Lorenzo V, Danchin A. Synthetic biology: discovering new worlds and new words. EMBO Rep. 2008; 9(9):822–7.Church GM, Elowitz MB, Smolke CD, Voigt CA, Weiss R. Realizing the potential of synthetic biology. Nat Rev Mol Cell Biol. 2014; 15(4):289–94.Takahashi CN, Miller AW, Ekness F, Dunham MJ, Klavins E. A low cost, customizable turbidostat for use in synthetic circuit characterization. ACS Synth Biol. 2015; 4(1):32–8. [doi: 10.1021/sb500165g ].Cooling MT, Rouilly V, Misirli G, Lawson J, Yu T, Hallinan J, Wipat A. Standard virtual biological parts: a repository of modular modeling components for synthetic biology. Bioinformatics. 2010; 26(7):925–31.Medema MH, van Raaphorst R, Takano E, Breitling R. Computational tools for the synthetic design of biochemical pathways. Nat Rev Microbiol. 2012; 10(3):191–202.Marchisio MA, Stelling J. Automatic design of digital synthetic gene circuits. PLoS Comput Biol. 2011; 7(2):e1001083. [doi: 10.1371/journal.pcbi.1001083 ].Rodrigo G, Carrera J, Landrain TE, Jaramillo A. Perspectives on the automatic design of regulatory systems for synthetic biology. FEBS Lett. 2012; 586(15):2037–42.Crook N, Alper HS. Model-based design of synthetic, biological systems. Chem Eng Sci. 2013; 103:2–11.Jayanthi S, Nilgiriwala K, Del Vecchio D. Retroactivity controls the temporal dynamics of gene transcription. ACS Synth Biol. 2013; 2(8):431–41.Mélykúti B, Hespanha JP, Khammash M. Equilibrium distributions of simple biochemical reaction systems for time-scale separation in stochastic reaction networks. J R Soc Interface. 2014; 11(97):20140054.Oyarzún DA, Lugagne JB, Stan GB. Noise propagation in synthetic gene circuits for metabolic control. ACS Synth Biol. 2015; 4(2):116–25. [doi: 10.1021/sb400126a ].Picó J, Vignoni A, Picó-Marco E, Boada Y. Modelado de sistemas bioquímicos: De la ley de acción de masas a la aproximación lineal del ruido. Revista Iberoamericana de Automática e Informática Industrial RIAI. 2015; 12(3):241–52.Feng X-j-J, Hooshangi S, Chen D, Li G, Weiss R, Rabitz H. Optimizing genetic circuits by global sensitivity analysis. Biophys J. 2004; 87(4):2195–202.Dasika MS, Maranas CD. Optcircuit: An optimization based method for computational design of genetic circuits. BMC Syst Biol. 2008; 2:24.Rodrigo G, Carrera J, Jaramillo A. Genetdes. Bioinformatics. 2007; 23(14):1857–8.Otero-Muras I, Banga JR. Multicriteria global optimization for biocircuit design. 2014. arXiv preprint arXiv:1402.7323.Banga JR. Optimization in computational systems biology. BMC Syst Biol. 2008; 2:47.Sendin J, Exler O, Banga JR. Multi-objective mixed integer strategy for the optimisation of biological networks. IET Syst Biol. 2010; 4(3):236–48.Miller M, Hafner M, Sontag E, Davidsohn N, Subramanian S, Purnick PE, Lauffenburger D, Weiss R. Modular design of artificial tissue homeostasis: robust control through synthetic cellular heterogeneity. PLoS Comput Biol. 2012; 8(7):1002579.Ellis T, Wang X, Collins JJ. Diversity-based, model-guided construction of synthetic gene networks with predicted functions. Nat Biotechnol. 2009; 27(5):465–71.Koeppl H, Hafner M, Lu J. Mapping behavioral specifications to model parameters in synthetic biology. BMC Bioinforma. 2013; 14(Suppl 10):9.Chiang AWT, Hwang M-JJ. A computational pipeline for identifying kinetic motifs to aid in the design and improvement of synthetic gene circuits. BMC Bioinforma. 2013; 14 Suppl 16:5.Ma W, Trusina A, El-Samad H, Lim WA, Tang C. Defining network topologies that can achieve biochemical adaptation. Cell. 2009; 138(4):760–73.Chiang AWT, Liu W-CC, Charusanti P, Hwang M-JJ. Understanding system dynamics of an adaptive enzyme network from globally profiled kinetic parameters. BMC Syst Biol. 2014; 8:4.Reynoso-Meza G, Blasco X, Sanchis J, Martínez M. Controller tuning using evolutionary multi-objective optimisation: current trends and applications. Control Eng Pract. 2014; 28:58–73.Alon U. An Introduction To Systems Biology. Design Principles of Biological Circuits. London: Chapman & Hall/ CRC Mathematical and computational Biology Series; 2006.Elowitz MB, Leibler S. A synthetic oscillatory network of transcriptional regulators. Nature. 2000; 403(6767):335–8.Hsiao V, de los Santos ELC, Whitaker WR, Dueber JE, Murray RM. Design and implementation of a biomolecular concentration tracker. ACS Synth Biol. 2015; 4(2):150–61. [doi: 10.1021/sb500024b ].Franco E, Giordano G, Forsberg P-O, Murray RM. Negative autoregulation matches production and demand in synthetic transcriptional networks. ACS Synth Biol. 2014; 3(8):589–99. [doi: 10.1021/sb400157z ].Strelkowa N, Barahona M. Switchable genetic oscillator operating in quasi-stable mode. J R Soc Interface. 2010; 7(48):1071–82.Basu S, Mehreja R, Thiberge S, Chen MT, Weiss R. Spatiotemporal control of gene expression with pulse-generating networks. Proc Natl Acad Sci U S A. 2004; 101(17):6355–60.Bleris L, Xie Z, Glass D, Adadey A, Sontag E, Benenson Y. Synthetic incoherent feedforward circuits show adaptation to the amount of their genetic template. Mol Syst Biol. 2011; 7(519):1–12. [doi: 10.1038/msb.2011.49 ].Hart Y, Antebi YE, Mayo AE, Friedman N, Alon U. Design principles of cell circuits with paradoxical components. Proc Natl Acad Sci. 2012; 109(21):8346–51.Zhang Q, Bhattacharya S, Andersen ME. Ultrasensitive response motifs: basic amplifiers in molecular signalling networks. Open Biol. 2013; 3(4):130031.Weber M, Buceta J, Others. Dynamics of the quorum sensing switch: stochastic and non-stationary effects. BMC Syst Biol. 2013; 7(1):6.Womelsdorf T, Valiante TA, Sahin NT, Miller KJ, Tiesinga P. Dynamic circuit motifs underlying rhythmic gain control, gating and integration. Nat Neurosci. 2014; 17(8):1031–9.Arpino JAJ, Hancock EJ, Anderson J, Barahona M, Stan G-BVB, Papachristodoulou A, Polizzi K. Tuning the dials of synthetic biology. Microbiology. 2013; 159(Pt 7):1236–53.Zagaris A, Kaper HGG, Kaper TJJ. Analysis of the computational singular perturbation reduction method for chemical kinetics. J Nonlinear Sci. 2004; 14(1):59–91.Anderson J, Chang Y-C-C, Papachristodoulou A. Model decomposition and reduction tools for large-scale networks in systems biology. Automatica. 2011; 47(6):1165–74.Prescott TP, Papachristodoulou A. Layered decomposition for the model order reduction of timescale separated biochemical reaction networks. J Theor Biol. 2014; 356:113–22.Hancock EJ, Stan GB, Arpino JAJ, Papachristodoulou A. Simplified mechanistic models of gene regulation for analysis and design. J R Soc Interface. 2015; 12(108).Miettinen K, Vol. 12. Nonlinear Multiobjective Optimization. Boston: Kluwer Academic Publishers; 1999.Miettinen K, Ruiz F, Wierzbicki AP. Introduction to multiobjective optimization: interactive approaches. In: Multiobjective Optimization. Berlin: Springer: 2008. p. 27–57.Deb K, Bandaru S, Greiner D, Gaspar-Cunha A, Tutum CC. An integrated approach to automated innovization for discovering useful design principles: Case studies from engineering. Appl Soft Comput. 2014; 15(0):42–56.Ang J, Ingalls B, McMillen D. Probing the input-output behavior of biochemical and genetic systems: System identification methods from control theory In: Johnson ML, Brand L, editors. Methods in Enzymology. Academic Press: 2011. p. 279–317, doi: 10.1016/B978-0-12-381270-4.00010-X .Mattson CA, Messac A. Pareto frontier based concept selection under uncertainty, with visualization. Optim Eng. 2005; 6(1):85–115.Reynoso-Meza G, Sanchis J, Blasco X, Martínez M. Design of continuous controllers using a multiobjective differential evolution algorithm with spherical pruning. Appl Evol Comput. 2010;532–541.Reynoso-Meza G, García-Nieto S, Sanchis J, Blasco X. Controller tuning using multiobjective optimization algorithms: a global tuning framework. IEEE Trans Control Syst Technol. 2013; 21(2):445–58.Reynoso-Meza G, Sanchis J, Blasco X, Herrero JM. Multiobjective evolutionary algortihms for multivariable PI controller tuning. Expert Syst Appl. 2012; 39:7895–907.Anderson C. Anderson promoter collection [online]. 2006. http://parts.igem.org/Promoters/Catalog/Anderson . Accesed 20 Feb 2015.Salis HM, Mirsky EA, Voigt CA. Automated design of synthetic ribosome binding sites to control protein expression. Nat Biotechnol. 2009; 27(10):946–50.Egbert RG, Klavins E. Fine-tuning gene networks using simple sequence repeats. PNAS. 2012; 109(42):16817–22. [doi: 10.1073/pnas.1205693109 ].Hair JF, Suárez MG. Análisis Multivariante vol. 491. Madrid: Prentice Hall; 1999.Blasco X, Herrero JM, Sanchis J, Martínez M. A new graphical visualization of n-dimensional pareto front for decision-making in multiobjective optimization. Inf Sci. 2008; 178(20):3908–24. [doi: 10.1016/j.ins.2008.06.010 ].Reynoso-Meza G, Blasco X, Sanchis J, Herrero JM. Comparison of design concepts in multi-criteria decision-making using level diagrams. Inform Sci. 2013; 221:124–41.Goentoro L, Shoval O, Kirschner MW, Alon U. The incoherent feedforward loop can provide fold-change detection in gene regulation. Mol Cell. 2009; 36(5):894–9.Rodrigo G, Elena SF. Structural discrimination of robustness in transcriptional feedforward loops for pattern formation. PloS ONE. 2011; 6(2):16904.Kim J, Khetarpal I, Sen S, Murray RM. Synthetic circuit for exact adaptation and fold-change detection. Nucleic Acids Res. 2014; 42(2):6078–89. [doi: 10.1093/nar/gku233 ].Chelliah V, Juty N, Ajmera I, Ali R, Dumousseau M, Glont M, Hucka M, Jalowicki G, Keating S, Knight-Schrijver V, et al. Biomodels: ten-year anniversary. Nucleic Acids Res. 2015; 43(D1):542–8.Ang J, Bagh S, Ingalls BP, McMillen DR. Considerations for using integral feedback control to construct a perfectly adapting synthetic gene network. J Theor Biol. 2010; 266(4):723–38.Biobrick Foundation. 2006. Part Registry [online]. http://partsregistry.org/ . Accessed 20 Feb 2015.BIOSS. 2006. BIOSS Toolbox [online]. http://www.bioss.uni-freiburg.de/cms/toolbox-home.html . Accessed 20 Feb 2015.BioFab. 2006. International Open Facility Advancing Biotechnology [online]. http://www.biofab.org/ . Accessed 20 Feb 2015.Vallerio M, Hufkens J, Van Impe J, Logist F. An interactive decision-support system for multi-objective optimization of nonlinear dynamic processes with uncertainty. Expert Syst Appl. 2015; 42(21):7710–31.Frangopol DM, Maute K. Life-cycle reliability-based optimization of civil and aerospace structures. Comput Struct. 2003; 81(7):397–410.Lozano M, Molina D, Herrera F. Editorial scalability of evolutionary algorithms and other metaheuristics for large-scale continuous optimization problems. Soft Comput. 2011; 15(11):2085–7.Santana-Quintero LV, Montano AA, Coello CAC. A review of techniques for handling expensive functions in evolutionary multi-objective optimization. In: Computational Intelligence in Expensive Optimization Problems. Berlin: Springer: 2010. p. 29–59

Fast stochastic simulation of biochemical reaction systems by alternative formulations of the Chemical Langevin Equation

The Chemical Langevin Equation (CLE), which is a stochastic differential equation (SDE) driven by a multidimensional Wiener process, acts as a bridge between the discrete Stochastic Simulation Algorithm and the deterministic reaction rate equation when simulating (bio)chemical kinetics. The CLE model is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. The contribution of this work is that we observe and explore that the CLE is not a single equation, but a parametric family of equations, all of which give the same finite-dimensional distribution of the variables. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation. On the practical side, we show that in the case where there are m1 pairs of reversible reactions and m2 irreversible reactions only m1+m2 Wiener processes are required in the formulation of the CLE, whereas the standard approach uses 2m1 + m2. We illustrate our findings by considering alternative formulations of the CLE for a HERG ion channel model and the Goldbeter–Koshland switch. We show that there are considerable computational savings when using our insights

A multi-time-scale analysis of chemical reaction networks: II. Stochastic systems

We consider stochastic descriptions of chemical reaction networks in which there are both fast and slow reactions, and for which the time scales are widely separated. We develop a computational algorithm that produces the generator of the full chemical master equation for arbitrary systems, and show how to obtain a reduced equation that governs the evolution on the slow time scale. This is done by applying a state space decomposition to the full equation that leads to the reduced dynamics in terms of certain projections and the invariant distributions of the fast system. The rates or propensities of the reduced system are shown to be the rates of the slow reactions conditioned on the expectations of fast steps. We also show that the generator of the reduced system is a Markov generator, and we present an efficient stochastic simulation algorithm for the slow time scale dynamics. We illustrate the numerical accuracy of the approximation by simulating several examples. Graph-theoretic techniques are used throughout to describe the structure of the reaction network and the state-space transitions accessible under the dynamics.clos

A higher-order numerical framework for stochastic simulation of chemical reaction systems

<p>Abstract</p> <p>Background</p> <p>In this paper, we present a framework for improving the accuracy of fixed-step methods for Monte Carlo simulation of discrete stochastic chemical kinetics. Stochasticity is ubiquitous in many areas of cell biology, for example in gene regulation, biochemical cascades and cell-cell interaction. However most discrete stochastic simulation techniques are slow. We apply Richardson extrapolation to the moments of three fixed-step methods, the Euler, midpoint and <it>θ</it>-trapezoidal <it>τ</it>-leap methods, to demonstrate the power of stochastic extrapolation. The extrapolation framework can increase the order of convergence of any fixed-step discrete stochastic solver and is very easy to implement; the only condition for its use is knowledge of the appropriate terms of the global error expansion of the solver in terms of its stepsize. In practical terms, a higher-order method with a larger stepsize can achieve the same level of accuracy as a lower-order method with a smaller one, potentially reducing the computational time of the system.</p> <p>Results</p> <p>By obtaining a global error expansion for a general weak first-order method, we prove that extrapolation can increase the weak order of convergence for the moments of the Euler and the midpoint <it>τ</it>-leap methods, from one to two. This is supported by numerical simulations of several chemical systems of biological importance using the Euler, midpoint and <it>θ</it>-trapezoidal <it>τ</it>-leap methods. In almost all cases, extrapolation results in an improvement of accuracy. As in the case of ordinary and stochastic differential equations, extrapolation can be repeated to obtain even higher-order approximations.</p> <p>Conclusions</p> <p>Extrapolation is a general framework for increasing the order of accuracy of any fixed-step stochastic solver. This enables the simulation of complicated systems in less time, allowing for more realistic biochemical problems to be solved.</p