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Stochastic simulation algorithms for computational systems biology: Exact, approximate, and hybrid methods

Advanced Review

Giulia Simoni, Federico Reali, Corrado Priami, Luca Marchetti

Published Online: Jul 01 2019

DOI: 10.1002/wsbm.1459

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Abstract Nowadays, mathematical modeling is playing a key role in many different research fields. In the context of system biology, mathematical models and their associated computer simulations constitute essential tools of investigation. Among the others, they provide a way to systematically analyze systems perturbations, develop hypotheses to guide the design of new experimental tests, and ultimately assess the suitability of specific molecules as novel therapeutic targets. To these purposes, stochastic simulation algorithms (SSAs) have been introduced for numerically simulating the time evolution of a well‐stirred chemically reacting system by taking proper account of the randomness inherent in such a system. In this work, we review the main SSAs that have been introduced in the context of exact, approximate, and hybrid stochastic simulation. Specifically, we will introduce the direct method (DM), the first reaction method (FRM), the next reaction method (NRM) and the rejection‐based SSA (RSSA) in the area of exact stochastic simulation. We will then present the τ‐leaping method and the chemical Langevin method in the area of approximate stochastic simulation and an implementation of the hybrid RSSA (HRSSA) in the context of hybrid stochastic‐deterministic simulation. Finally, we will consider the model of the sphingolipid metabolism to provide an example of application of SSA to computational system biology by exemplifying how different simulation strategies may unveil different insights into the investigated biological phenomenon. This article is categorized under: Models of Systems Properties and Processes > Mechanistic Models Analytical and Computational Methods > Computational Methods

A graphical representation of the simulation algorithms introduced in the review. Starting from a common root node representing a generic stochastic simulation algorithm, the methodologies differentiate in terms of accuracy and runtime according to exact and approximate methods. We depict with rectangles the algorithm classes and with circles the specific methods. Hybrid methods are here represented as a part of the approximate methods, however, they are often referred as a class of simulation algorithms itself. In the diagram, the τ‐leaping and the chemical Langevin methods are connected since, following the Gillespie's approach, the latter can be derived as an approximation of the former. Analogously, the deterministic methods are connected with the chemical Langevin method. Deterministic methods are indicated with dashed lines since they are not described in detail in this review

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Distribution of the number of molecules for the species ceramide‐1‐phosphate (Cer1P) and sphingosine‐1‐phosphate (S1P) at the end of the simulation (t = 24 hr) using rejection‐based SSA (RSSA) and the RSSA version that accounts for extrinsic noise (10,000 simulation runs)

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The comparison of the simulation results of Reali et al. model using a deterministic approach and rejection‐based SSA (RSSA) in combination with a scaling factor of 1e‐03 to reduce the population abundance. The results highlight how the stochastic noise is affecting the less abundant species ceramide‐1‐phosphate (Cer1P) and sphingosine‐1‐phosphate (S1P), suggesting the use of exact or hybrid stochastic simulation algorithms

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Schematic overview of a hybrid simulation strategy in which the set of reactions is divided into two subsets of slow and fast reactions. The two subsets are simulated with different methods and then are synchronized in order to define the new system state

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Two outcomes of the simulation of the Lotka‐Volterra model with different simulation algorithms. Figure a Shows the results of a deterministic simulation dynamics. Figure b Shows the dynamics of the same model using a stochastic algorithm. x‐axis: Time in absolute units. y‐axis: Abundance of the species

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References

Ahn,, T.‐H., Han,, X., & Sandu,, A. (2015). Implicit simulation methods for stochastic chemical kinetics. Journal of Applied Analysis and Computation, 5(3), 420–452. https://doi.org/10.11948/2015034
Ahn,, T.‐H., & Sandu,, A. (2011a). Fully implicit tau‐leaping methods for the stochastic simulation of chemical kinetics. In Proceedings of the 19th high performane computing symposia (pp. 118–125), Boston, MA.
Ahn,, T.‐H., & Sandu,, A. (2011b). Implicit second order weak Taylor tau‐leaping methods for the stochastic simulation of chemical kinetics. Procedia Computer Science, 4, 2297–2306. https://doi.org/10.1016/j.procs.2011.04.250
Alfonsi,, A., Cancès,, E., Turinici,, G., Di Ventura,, B., & Huisinga,, W. (2005). Adaptive simulation of hybrid stochastic and deterministic models for biochemical systems. ESAIM: Proceedings, 14, 1–13. https://doi.org/10.1051/proc:2005001
Alvarez‐Vasquez,, F., Sims,, K. J., Cowart,, L. A., Okamoto,, Y., Voit,, E. O., & Hannun,, Y. A. (2005). Simulation and validation of modelled sphingolipid metabolism in Saccharomyces cerevisiae. Nature, 433, 425–430. https://doi.org/10.1038/nature03232
Anderson,, D. F. (2007). A modified next reaction method for simulating chemical systems with time dependent propensities and delays. The Journal of Chemical Physics, 127(21), 214107. https://doi.org/10.1063/1.2799998
Anderson,, D. F. (2008). Incorporating postleap checks in tau‐leaping. The Journal of Chemical Physics, 128(5), 054103. https://doi.org/10.1063/1.2819665
Anderson,, D. F., Higham,, D. J., & Sun,, Y. (2016). Multilevel Monte Carlo for stochastic differential equations with additive small noise. SIAM Journal on Numerical Analysis, 54(2), 505–529. https://doi.org/10.1007/s10479-009-0663-8
Auger,, A., Chatelain,, P., & Koumoutsakos,, P. (2006). R‐leaping: Accelerating the stochastic simulation algorithm by reaction leaps. The Journal of Chemical Physics, 125(8), 084103. https://doi.org/10.1063/1.2218339
Baker,, R. E., Yates,, C. A., & Erban,, R. (2010). From microscopic to macroscopic descriptions of cell migration on growing domains. Bulletin of Mathematical Biology, 72(3), 719–762.
Bentele,, M., & Eils,, R. (2005). General stochastic hybrid method for the simulation of chemical reaction processes in cells. In Computational methods in systems biology (pp. 248–251).
Burden,, R. L., Faires,, D. J., & Burden,, A. M. (2016). Numerical analysis (10th ed.). Boston, MA: Cengage Learning.
Butcher,, J. C. (2008). Numerical methods for ordinary differential equations (2nd ed.). Hoboken, NJ: Wiley.
Cai,, X., & Wang,, X. (2007). Stochastic modeling and simulation of gene networks‐a review of the state‐of‐the‐art research on stochastic simulations. IEEE Signal Processing Magazine, 24(1), 27–36.
Cai,, X., & Xu,, Z. (2007). K‐leap method for accelerating stochastic simulation of coupled chemical reactions. The Journal of Chemical Physics, 126(7), 074102. https://doi.org/10.1063/1.2436869
Cao,, Y., Gillespie,, D. T., & Petzold,, L. R. (2005). Avoiding negative populations in explicit Poisson tau‐leaping. The Journal of Chemical Physics, 123(5), 054104. https://doi.org/10.1063/1.1992473
Cao,, Y., Gillespie,, D. T., & Petzold,, L. R. (2006). Efficient step size selection for the tau‐leaping simulation method. The Journal of Chemical Physics, 124(5), 044109. https://doi.org/10.1063/1.2159468
Cao,, Y., Gillespie,, D. T., & Petzold,, L. R. (2007). Adaptive explicit‐implicit tau‐leaping method with automatic tau selection. The Journal of Chemical Physics, 126(22), 224101. https://doi.org/10.1063/1.2745299
Cao,, Y., Li,, H., & Petzold,, L. R. (2004). Efficient formulation of the stochastic simulation algorithm for chemically reacting systems. The Journal of Chemical Physics, 121(9), 4059–4067. https://doi.org/10.1063/1.1778376
Cao,, Y., & Petzold,, L. R. (2005). Trapezoidal tau‐leaping formula for the stochastic simulation of biochemical systems. In Proceedings of foundations of systems biology in engineering (FOSBE 2005) (pp. 149–152), Santa Barbara, CA.
Chatterjee,, A., Mayawala,, K., Edwards,, J. S., & Vlachos,, D. G. (2005). Time accelerated Monte Carlo simulations of biological networks using the binomial τ‐leap method. Bioinformatics, 21(9), 2136–2137. https://doi.org/10.1093/bioinformatics/bti308
Chatterjee,, A., Vlachos,, D. G., & Katsoulakis,, M. A. (2005). Binomial distribution based τ‐leap accelerated stochastic simulation. The Journal of Chemical Physics, 122(2), 024412. https://doi.org/10.1063/1.1833357
Elowitz,, M. B., Levine,, A. J., Siggia,, E. D., & Swain,, P. S. (2002). Stochastic gene expression in a single cell. Science, 297(5584), 1183–1186. https://doi.org/10.1126/science.1070919
Ethier,, S. N., & Kurtz,, T. G. (1986). Markov processes: Characterization and. Convergence, 43, 484. https://doi.org/10.2307/2531839
Fisher,, J., & Henzinger,, T. A. (2007). Executable cell biology. Nature Biotechnology, 25(11), 1239–1249.
Gibson,, M. A., & Bruck,, J. (2000). Efficient exact stochastic simulation of chemical systems with many species and many channels. The Journal of Physical Chemistry A, 104(9), 1876–1889. https://doi.org/10.1021/jp993732q
Gillespie,, D. T. (1976). A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. Journal of Computational Physics, 22(4), 403–434. https://doi.org/10.1016/0021-9991(76)90041-3
Gillespie,, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry, 81(25), 2340–2361.
Gillespie,, D. T. (2000). The chemical Langevin equation. The Journal of Chemical Physics, 113(1), 297–306. https://doi.org/10.1063/1.481811
Gillespie,, D. T. (2001). Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, 115(4), 1716–1733. https://doi.org/10.1063/1.1378322
Gillespie,, D. T. (2002). The chemical Langevin and Fokker‐Planck equations for the reversible isomerization reaction. The Journal of Physical Chemistry A, 106(20), 5063–5071. https://doi.org/10.1021/jp0128832
Gillespie,, D. T. (2009). Deterministic limit of stochastic chemical kinetics. The Journal of Physical Chemistry B, 113(6), 1640–1644. https://doi.org/10.1021/jp806431b
Gillespie,, D. T., & Petzold,, L. R. (2003). Improved lead‐size selection for accelerated stochastic simulation. The Journal of Chemical Physics, 119(16), 8229–8234. https://doi.org/10.1063/1.1613254
Griffith,, M., Courtney,, T., Peccoud,, J., & Sanders,, W. H. (2006). Dynamic partitioning for hybrid simulation of the bistable HIV‐1 transactivation network. Bioinformatics, 22(22), 2782–2789. https://doi.org/10.1093/bioinformatics/btl465
Gupta,, S., Maurya,, M. R., Merrill,, A. H., Glass,, C. K., & Subramaniam,, S. (2011). Integration of lipidomics and transcriptomics data towards a systems biology model of sphingolipid metabolism. BMC Systems Biology, 5(1), 26. https://doi.org/10.1186/1752-0509-5-26
Halasz,, A., Kumar,, V., Imieliński,, M., Belta,, C., Sokolsky,, O., Pathak,, S., & Rubin,, H. (2007). Analysis of lactose metabolism in E. coli using reachability analysis of hybrid systems. IET Systems Biology, 1(2), 130–148.
Harris,, L. A., & Clancy,, P. (2006). A partitioned leaping approach for multiscale modeling of chemical reaction dynamics. The Journal of Chemical Physics, 125(14), 144107. https://doi.org/10.1063/1.2354085
Haseltine,, E. L., & Rawlings,, J. B. (2002). Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. The Journal of Chemical Physics, 117(15), 6959–6969. https://doi.org/10.1063/1.1505860
Hepp,, B., Gupta,, A., & Khammash,, M. (2015). Adaptive hybrid simulations for multiscale stochastic reaction networks. The Journal of Chemical Physics, 142(3), 034118. https://doi.org/10.1063/1.4905196
Ilie,, S. (2012). Variable time‐stepping in the pathwise numerical solution of the chemical Langevin equation. The Journal of Chemical Physics, 137(23), 234110. https://doi.org/10.1063/1.4771660
Ilie,, S., & Teslya,, A. (2012). An adaptive stepsize method for the chemical Langevin equation. The Journal of Chemical Physics, 136(18), 184101. https://doi.org/10.1063/1.4711143
Ilie,, S., & Morshed,, M. (2013). Automatic simulation of the chemical Langevin equation. Applied Mathematics, 4(1A), 235–241. https://doi.org/10.4236/am.2013.41A036
Irizarry,, R. (2011). Stochastic simulation of population balance models with disparate time scales: Hybrid strategies. Chemical Engineering Science, 66(18), 4059–4069. https://doi.org/10.1016/j.ces.2011.05.035
Kaddi, C., Niesner, B., Baek, R., Jasper, P., Pappas, J., Tolsma, J., … Azer, K. (2018). Quantitative systems pharmacology modeling of acid sphingomyelinase deficiency and the enzyme replacement therapy olipudase alfa is an innovative tool for linking pathophysiology and pharmacology. CPT Pharmacometrics %26 Systems Pharmacology, 7, 442–452.
Kaddi,, C., Reali,, F., Marchetti,, L., Niesner,, B., Parolo,, S., Simoni,, G., … Azer,, K. (2018). Integrated quantitative systems pharmacology (QSP) model of lysosomal diseases provides an innovative computational platform to support research and therapeutic development for the sphingolipidoses. Molecular Genetics and Metabolism Systems Biology, 123, S73–S74.
Kang,, H.‐W., & Kurtz,, T. G. (2013). Separation of time‐scales and model reduction for stochastic reaction networks. The Annals of Applied Probability, 23(2), 529–583. https://doi.org/10.1214/12-AAP841
Kurtz,, T. G. (1976). Limit theorems and diffusion approximations for density dependent markov chains. In R. J.‐B. Wets, (Ed.), Stochastic systems: Modeling, identification and optimization, i (pp. 67–78). Berlin, Heidelberg: Springer. https://doi.org/10.1007/BFb0120765
Kurtz,, T. G. (1978). Strong approximation theorems for density dependent Markov chains. Stochastic Processes and their Applications, 6(3), 223–240. https://doi.org/10.1016/0304-4149(78)90020-0
Li,, H., & Petzold,, L. R. (2006). Logarithmic direct method for discrete stochastic simulation of chemically reacting systems. Journal of Chemical Physics, 16, 1–11.
Li,, Y., & Hu,, L. (2015). A fast exact simulation algorithm for a class of markov jump processes. The Journal of Chemical Physics, 143(18), 184105. https://doi.org/10.1063/1.4934972
Lok,, L., & Brent,, R. (2005). Automatic generation of cellular reaction networks with Moleculizer 1.0. Nature Biotechnology, 23(1), 131–136. https://doi.org/10.1038/nbt1054
Lotka,, A. J. (1920). Analytical note on certain rhythmic relations in organic systems. Proceedings of the National Academy of Sciences, 6, 410–415.
Marchetti,, L., Priami,, C., & Thanh,, V. H. (2016). HRSSA—Efficient hybrid stochastic simulation for spatially homogeneous biochemical reaction networks. Journal of Computational Physics, 317, 301–317. https://doi.org/10.1016/j.jcp.2016.04.056
Marchetti,, L., Priami,, C., & Thanh,, V. H. (2017). Simulation algorithms for computational systems biology. Cham, Switzerland: Springer‐Verlag New York Inc.
Mauch,, S., & Stalzer,, M. (2011). Efficient formulations for exact stochastic simulation of chemical systems. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 8(1), 27–35. https://doi.org/10.1109/TCBB.2009.47
McAdams,, H. H., & Arkin,, A. (1999). It`s a noisy business! Genetic regulation at the nanomolar scale. Trends in Genetics, 15(2), 65–69. https://doi.org/10.1016/S0168-9525(98)01659-X
McCollum,, J. M., Peterson,, G. D., Cox,, C. D., Simpson,, M. L., & Samatova,, N. F. (2006). The sorting direct method for stochastic simulation of biochemical systems with varying reaction execution behavior. Computational Biology and Chemistry, 30(1), 39–49. https://doi.org/10.1016/j.compbiolchem.2005.10.007
Merrill,, A. H. (2011). Sphingolipid and glycosphingolipid metabolic pathways in the era of sphingolipidomics. Chemical Reviews, 111(10), 6387–6422. https://doi.org/10.1021/cr2002917
Misselbeck,, K., Marchetti,, L., Field,, M. S., Scotti,, M., Priami,, C., & Stover,, P. J. (2017). A hybrid stochastic model of folate‐mediated one‐carbon metabolism: Effect of the common c677t mthfr variant on de novo thymidylate biosynthesis. Scientific Reports, 7(1), 797.
Misselbeck,, K., Marchetti,, L., Priami,, C., Stover,, P. J., & Field,, M. S. (2019). The 5‐formyltetrahydrofolate futile cycle reduces pathway stochasticity in an extended hybrid‐stochastic model of folate‐mediated one‐carbon metabolism. Scientific Reports, 9(1), 4322.
Moraes,, A., Tempone,, R., & Vilanova,, P. (2014). Hybrid chernoff tau‐leap. Multiscale Modeling and Simulation, 12(2), 581–615.
Moraes,, A., Tempone,, R., & Vilanova,, P. (2015). Multilevel hybrid chernoff tau‐leap. BIT Numerical Mathematics, 56(1), 189–239. https://doi.org/10.1007/s10543-015-0556-y
Neogi,, N. A. (2004). Dynamic partitioning of large discrete event biological systems for hybrid simulation and analysis. Berlin and Heidelberg, Germany: Springer.
Pahle,, J. (2008). Biochemical simulations: Stochastic, approximate stochastic and hybrid approaches. Briefings in Bioinformatics, 10(1), 53–64. https://doi.org/10.1093/bib/bbn050
Peng,, X.‐J., & Wang,, Y.‐F. (2007). L‐leap: Accelerating the stochastic simulation of chemically reacting systems. Applied Mathematics and Mechanics, 28(10), 1361–1371. https://doi.org/10.1007/s10483-007-1009-y
Platt,, F. M. (2014). Sphingolipid lysosomal storage disorders. Nature, 510, 68–75.
Press,, W. H., Teukolsky,, S. A., Vetterling,, W. T., & Flannery,, B. P. (2007). Numerical recipes 3rd edition: The art of scientific computing. Cambridge, England: Cambridge University Press.
Priami,, C., & Morine,, M. J. (2015). Analysis of biological systems. London, England: Imperial College Press.
Quarteroni,, A., Sacco,, R., & Salieri,, F. (2007). Numerical mathematics. New York, NY: Springer.
Ramaswamy,, R., González‐Segredo,, N., & Sbalzarini,, I. F. (2009). A new class of highly efficient exact stochastic simulation algorithms for chemical reaction networks. The Journal of Chemical Physics, 130(24), 244104. https://doi.org/10.1063/1.3154624
Ramaswamy,, R., & Sbalzarini,, I. F. (2010). A partial‐propensity variant of the composition‐rejection stochastic simulation algorithm for chemical reaction networks. The Journal of Chemical Physics, 132(4), 044102. https://doi.org/10.1063/1.3297948
Rathinam,, M., Petzold,, L. R., Cao,, Y., & Gillespie,, D. T. (2003). Stiffness in stochastic chemically reacting systems: The implicit tau‐leaping method. The Journal of Chemical Physics, 119(24), 12784–12794. https://doi.org/10.1063/1.1627296
Reali,, F., Morine,, M. J., Kahramanoggullari,, O., Raichur,, S., Schneider,, H.‐C., Crowther,, D., & Priami,, C. (2017). Mechanistic interplay between ceramide and insulin resistance. Scientific Reports, 7, 41231.
Reali,, F., Priami,, C., & Marchetti,, L. (2017). Optimization algorithms for computational systems biology. Frontiers in Applied Mathematics and Statistics, 3, 6. http://doi.org/10.3389/fams.2017.00006
Salis,, H., & Kaznessis,, Y. (2005). Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions. The Journal of Chemical Physics, 122(5), 054103. https://doi.org/10.1063/1.1835951
Sandmann,, W. (2009). Streamlined formulation of adaptive explicit‐implicit tau‐leaping with automatic tau selection. In Proceedings of winter simulation conference (pp. 1104–1112). https://doi.org/10.1109/WSC.2009.5429309
Sanft,, K. R., & Othmer,, H. G. (2015). Constant‐complexity stochastic simulation algorithm with optimal binning. The Journal of Chemical Physics, 143(7), 074108. https://doi.org/10.1063/1.4928635
Santillán,, M., & Mackey,, M. C. (2008). Quantitative approaches to the study of bistability in the lac operon of Escherichia coli. Journal of the Royal Society Interface, 5(suppl_1), S29–S39.
Schulze,, T. (2008). Efficient kinetic Monte Carlo simulation. Journal of Computational Physics, 227(4), 2455–2462. https://doi.org/10.1016/j.jcp.2007.10.021
Slepoy,, A., Thompson,, A. P., & Plimpton,, S. J. (2008). A constant‐time kinetic Monte Carlo algorithm for simulation of large biochemical reaction networks. The Journal of Chemical Physics, 128(20), 205101. https://doi.org/10.1063/1.2919546
Swain,, P. S., Elowitz,, M. B., & Siggia,, E. D. (2002). Intrinsic and extrinsic contributions to stochasticity in gene expression. Proceedings of the National Academy of Sciences, 99(20), 12795–12800. https://doi.org/10.1073/pnas.162041399
Thanh,, V. H., Marchetti,, L., Reali,, F., & Priami,, C. (2018). Incorporating extrinsic noise into the stochastic simulation of biochemical reactions: A comparison of approaches. The Journal of Chemical Physics, 148(6), 064111. https://doi.org/10.1063/1.5016338
Thanh,, V. H., & Priami,, C. (2015). Simulation of biochemical reactions with time‐dependent rates by the rejection‐based algorithm. The Journal of Chemical Physics, 143(5), 054104. https://doi.org/10.1063/1.4927916
Thanh,, V. H., Priami,, C., & Zunino,, R. (2014). Efficient rejection‐based simulation of biochemical reactions with stochastic noise and delays. The Journal of Chemical Physics, 141(13), 134116. https://doi.org/10.1063/1.4896985
Thanh,, V. H., & Zunino,, R. (2012). Tree‐based search for stochastic simulation algorithm. In Proceedings of the ACM symposium on applied computing (pp. 1415–1416). https://doi.org/10.1145/2245276.2232001
Thanh,, V. H., & Zunino,, R. (2014). Adaptive tree‐based search for stochastic simulation algorithm. International Journal of Computational Biology and Drug Design, 7(4), 341–357.
Thanh,, V. H., Zunino,, R., & Priami,, C. (2015). On the rejection‐based algorithm for simulation and analysis of large‐scale reaction networks. The Journal of Chemical Physics, 142(24), 244106. https://doi.org/10.1063/1.4922923
Thanh,, V. H., Zunino,, R., & Priami,, C. (2017a). Efficient constant‐time complexity algorithm for stochastic simulation of large reaction networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 14(3), 657–667. https://doi.org/10.1109/TCBB.2016.2530066
Thanh,, V. H., Zunino,, R., & Priami,, C. (2017b). Efficient stochastic simulation of biochemical reactions with noise and delays. The Journal of Chemical Physics, 146(8), 084107. https://doi.org/10.1063/1.4976703
Tian,, T., & Burrage,, K. (2004). Binomial leap methods for simulating stochastic chemical kinetics. The Journal of Chemical Physics, 121(21), 10356–10364. https://doi.org/10.1063/1.1810475
Volterra,, V. (1926). Fluctuations in the abundance of a species considered mathematically. Nature, 118, 558–560.
Wilkinson,, D. J. (2009). Stochastic modelling for quantitative description of heterogeneous biological systems. Nature Reviews Genetics, 10(2), 122–133.
Wronowska,, W., Charzyńska,, A., Nienałtowski,, K., & Gambin,, A. (2015). Computational modeling of sphingolipid metabolism. BMC Systems Biology, 9(1), 47. https://doi.org/10.1186/s12918-015-0176-9

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