細胞作為生化反應的微小單位，當中的生化系統會受到化學成分中不確定性，或稱作noise，的顯著影響。 Noise的控制有很多層次。有報導指出noise的控制不力或錯誤與疾病有關。 儘管數值模擬結果可以提供noise的樣貌，但結果取決於速率常數和初始條件等參數。對於大多數生物系統來說，這些參數通常是未知的。例如分析擾動的響應時，測試每個參數的擾動結果，並顧及不同的參數組合的可能性，是非常耗時的。此外，當前的理論和軟體並未明確定義noise對擾動的響應。因此，一個能從生化網絡結構獲得總體的noise響應，以及一個結合了不同noise分析技術的軟體程式，都是當前非常需要的。
我們擴展了先前開發的law of localization (Okada, T. and Mochizuki, A.; Phys. Rev. Lett.; 117, 2016, 048101) 理論，用於預測化學物質的方差和協方差在擾動下的變化，據此開發了noise localization理論。 通過linear noise approximation，我們可以將生化網絡的微分方程擴展，使其包含方差和協方差的微分方程，組合成為一個虛擬網絡。 通過localization分析，可以獲得方差和協方差等“虛擬成分”在穩態下的擾動響應，寫成僅需要網絡拓撲的靈敏度矩陣。 我們的擴展理論可以在物種、方差和協方差級別識別緩衝結構，並且可以以類似化學反應的形式提供對非穩態條件下noise flow的洞察。 我們的理論適用於判別模型、掃描具有有趣noise行為的網絡拓撲，以及設計和擾動具有所需響應的網絡。
我們還開發了一個名為“BioSANS”的軟體，用於生化反應系統的解析數學求解和數值模擬，以及各種類型的noise分析。 它的特性包括了簡單的輸入格式、解析數學或數值計算、參數估計、圖形用戶界面，並且支持系統生物學標記語言 (SBML)。 對於生化系統中的動力學研究，BioSANS 提供了一套高度可用的全面性的工具。

Cells as tiny compartments for biochemical reactions, and biochemical systems can be significantly impacted by the uncertainties, or noises, in chemical composition. There are many levels of noise regulation and poor or erroneous regulation has been linked to diseases. Numerical simulations is a way to examine noise, but the results depend on parameters like rate constants and initial conditions which are typically unknown for most biological systems. For example, to analyze the response to perturbation, it is very inefficient to sample parameters and test for numerical perturbation of each parameter. Moreover, current theories and softwares do not explicitly define noise response to perturbation. It is highly desirable to have a general understanding of noise response based on network structure and a software that combines different noise analytical techniques.
We established noise localization theory by extending the previously developed law of localization (Okada, T. and Mochizuki, A.; Phys. Rev. Lett.; 117, 2016, 048101) for characterizing noise in terms of (co)variances. Using linear noise approximation, a biochemical network can be expanded into an extended set of differential equations, representing a fictitious network for pseudo-components consisting of variances and covariances as well as chemical species. By utilizing localization analysis, the responses to perturbation at steady state for the pseudo-components can be summarized from network structure information. Our research enables the identification of buffering structures at the extent of species, variance, and covariances and is capable of providing insights into noise flow under non-steady-state conditions from pseudo-chemical reactions. Noise localization could be used to discriminate models, scan network topologies with interesting noise behavior, and in the design and perturbation of networks with the desired response.
We also develop a software called “BioSANS” for symbolic mathematical solutions and numeric simulations for biochemical systems, as well as various types of noise analysis. It provides desirable features such as simplicity of input preparation, symbolic or numerical computation, parameter estimation, graphical user interface, and support for systems biology markup language (SBML). For studying kinetics and dynamics in biochemical systems, BioSANS provides a comprehensive collection of tools with improved availability and accessibility.

[1] M. B. Elowitz, A. J. Levine, E. D. Siggia, P. S. Swain, Science 2002, 297, 1183–1186.
[2] A. Hilfinger, J. Paulsson, PNAS 2011, 108, 12167–12172.
[3] P. S. Swain, M. B. Elowitz, E. D. Siggia, PNAS 2002, 99, 12795–12800.
[4] M. Kærn, T. C. Elston, W. J. Blake, J. J. Collins, Nat Rev Genet 2005, 6, 451–464.
[5] A. Klosin, F. Oltsch, T. Harmon, A. Honigmann, F. Jülicher, A. A. Hyman, C. Zechner, Science 2020, 367, 464–468.
[6] D. Gonze, J. Halloy, A. Goldbeter, Proc Natl Acad Sci U S A 2002, 99, 673–678.
[7] J. M. Schmiedel, S. L. Klemm, Y. Zheng, A. Sahay, N. Blüthgen, D. S. Marks, A. van Oudenaarden, Science 2015, 348, 128–132.
[8] G. Hornung, N. Barkai, PLoS Comput Biol 2008, 4, e8.
[9] I. Lestas, G. Vinnicombe, J. Paulsson, Nature 2010, 467, 174–178.
[10] M. T. Guinn, G. Balázsi, Nucleic Acids Research 2019, 47, 7703–7714.
[11] Y. Pilpel, in Yeast Systems Biology: Methods and Protocols (Eds.: J.I. Castrillo, S.G. Oliver), Humana Press, Totowa, NJ, 2011, pp. 407–425.
[12] C. H. Ko, Y. R. Yamada, D. K. Welsh, E. D. Buhr, A. C. Liu, E. E. Zhang, M. R. Ralph, S. A. Kay, D. B. Forger, J. S. Takahashi, PLOS Biology 2010, 8, e1000513.
[13] H. Safdari, A. Kalirad, C. Picioreanu, R. Tusserkani, B. Goliaei, M. Sadeghi, PLOS ONE 2020, 15, e0232060.
[14] T. Wada, S. Wallerich, A. Becskei, Cell Reports 2018, 24, 3503–3512.
[15] S. Bessonnard, L. De Mot, D. Gonze, M. Barriol, C. Dennis, A. Goldbeter, G. Dupont, C. Chazaud, Development 2014, 141, 3637–3648.
[16] M. Richard, G. Yvert, Frontiers in Genetics 2014, 5.
[17] D. A. McQuarrie, Journal of Applied Probability 1967, 4, 413–478.
[18] D. T. Gillespie, Physica A: Statistical Mechanics and its Applications 1992, 188, 404–425.
[19] T. Jahnke, W. Huisinga, J. Math. Biol. 2007, 54, 1–26.
[20] D. T. Gillespie, J. Phys. Chem. 1977, 81, 2340–2361.
[21] Y. Cao, D. T. Gillespie, L. R. Petzold, The Journal of Chemical Physics 2006, 124, 044109.
[22] D. T. Gillespie, The Journal of Chemical Physics 2000, 113, 297–306.
[23] T. G. Kurtz, in Stochastic Systems: Modeling, Identification and Optimization, I (Ed.: R.J.-B. Wets), Springer Berlin Heidelberg, Berlin, Heidelberg, 1976, pp. 67–78.
[24] J. Elf, M. Ehrenberg, Genome Res 2003, 13, 2475–2484.
[25] A. Mochizuki, B. Fiedler, Journal of Theoretical Biology 2015, 367, 189–202.
[26] T. Okada, A. Mochizuki, Phys. Rev. Lett. 2016, 117, 048101.
[27] N. Rosenfeld, J. W. Young, U. Alon, P. S. Swain, M. B. Elowitz, Science 2005, 307, 1962–1965.
[28] J. Ou, C. Furusawa, T. Yomo, H. Shimizu, Biosystems 2009, 97, 160–164.
[29] E. van Nimwegen, 2016, 049486.
[30] A. Singh, M. Soltani, PLOS ONE 2013, 8, e84301.
[31] A. Baudrimont, V. Jaquet, S. Wallerich, S. Voegeli, A. Becskei, Cell Reports 2019, 26, 3752-3761.e5.
[32] B.-R. Chen, C.-X. You, C.-C. Shu, Biosystems 2020, 198, 104269.
[33] J. K. Kim, A. A. Kolodziejczyk, T. Ilicic, S. A. Teichmann, J. C. Marioni, Nat Commun 2015, 6, 8687.
[34] P. Brennecke, S. Anders, J. K. Kim, A. A. Kołodziejczyk, X. Zhang, V. Proserpio, B. Baying, V. Benes, S. A. Teichmann, J. C. Marioni, M. G. Heisler, Nat Methods 2013, 10, 1093–1095.
[35] D. Kulasiri, R. Kosarwal, Chemical Master Equation for Large Biological Networks: State-Space Expansion Methods Using AI, Springer, Singapore, 2021.
[36] G. Altan-Bonnet, A. Libchaber, O. Krichevsky, Phys. Rev. Lett. 2003, 90, 138101.
[37] H. Y. Wang, T. Elston, A. Mogilner, G. Oster, Biophys J 1998, 74, 1186–1202.
[38] J. Paulsson, M. Ehrenberg, Phys Rev Lett 2000, 84, 5447–5450.
[39] J. Gorecki, A. L. Kawczyński, B. Nowakowski, J. Phys. Chem. A 1999, 103, 3200–3209.
[40] B. Nowakowski, J. Gorecki, A. L. Kawczyński, J. Phys. Chem. A 1998, 102, 7250–7256.
[41] M. Reis, J. A. Kromer, E. Klipp, J. Math. Biol. 2018, 77, 377–419.
[42] Y. Georgievskii, J. A. Miller, M. P. Burke, S. J. Klippenstein, J. Phys. Chem. A 2013, 117, 12146–12154.
[43] G. P. Basharin, A. N. Langville, V. A. Naumov, Linear Algebra and its Applications 2004, 386, 3–26.
[44] M. J. Thomas, U. Klein, J. Lygeros, M. Rodríguez Martínez, Frontiers in Immunology 2019, 10.
[45] R. Gui, Q. Liu, Y. Yao, H. Deng, C. Ma, Y. Jia, M. Yi, Frontiers in Physiology 2016, 7.
[46] K. M. Kamina, S. Mwalili, A. Wanjoya, International Journal of Data Science and Analysis 2019, 5, 117.
[47] E. J. Hancock, J. Ang, A. Papachristodoulou, G.-B. Stan, Cell Systems 2017, 5, 498-508.e23.
[48] D. Hnisz, K. Shrinivas, R. A. Young, A. K. Chakraborty, P. A. Sharp, Cell 2017, 169, 13–23.
[49] J. Melunis, U. Hershberg, Engineering Applications of Artificial Intelligence 2017, 62, 304–311.
[50] R. Wang, in Encyclopedia of Systems Biology (Eds.: W. Dubitzky, O. Wolkenhauer, K.-H. Cho, H. Yokota), Springer, New York, NY, 2013, pp. 1436–1437.
[51] D. F. Anderson, The Journal of Chemical Physics 2007, 127, 214107.
[52] M. A. Gibson, J. Bruck, J. Phys. Chem. A 2000, 104, 1876–1889.
[53] A. Slepoy, A. P. Thompson, S. J. Plimpton, J. Chem. Phys. 2008, 128, 205101.
[54] D. T. Gillespie, J. Chem. Phys. 2001, 115, 1716–1733.
[55] S. Wu, J. Fu, L. R. Petzold, J Chem Phys 2015, 142, DOI 10.1063/1.4921638.
[56] J. M. A. Padgett, S. Ilie, AIP Advances 2016, 6, 035217.
[57] Z. Xu, X. Cai, J. Chem. Phys. 2008, 128.
[58] P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer Berlin Heidelberg, Berlin, Heidelberg, 1992.
[59] A. Ceccato, D. Frezzato, J. Chem. Phys. 2018, 148, 064114.
[60] V. Sotiropoulos, M.-N. Contou-Carrere, P. Daoutidis, Y. N. Kaznessis, 2009, 6, 13.
[61] D. T. Gillespie, J. Phys. Chem. A 2002, 106, 5063–5071.
[62] C.-C. S. Yan, S. R. Chepyala, C.-M. Yen, C.-P. Hsu, Scientific Reports 2017, 7, 16851.
[63] X. Han, M. Valorani, H. N. Najm, J. Chem. Phys. 2019, 150, 194101.
[64] F. Wu, T. Tian, J. B. Rawlings, G. Yin, J. Chem. Phys. 2016, 144, 174112.
[65] T. Li, J. Phys. Chem. A 2020, 124, 810–816.
[66] N. G. V. Kampen, Stochastic Processes in Physics and Chemistry, North Holland, Amsterdam ; Boston, 2007.
[67] F. Hayot, C. Jayaprakash, Phys. Biol. 2004, 1, 205–210.
[68] P. Lötstedt, Bull Math Biol 2019, 81, 2873–2901.
[69] J. Fintzi, J. Wakefield, V. N. Minin, Biometrics 2022, 78, 1530–1541.
[70] P. Fearnhead, V. Giagos, C. Sherlock, Biometrics 2014, 70, 457–466.
[71] M. Komorowski, J. Miękisz, M. P. H. Stumpf, Biophysical Journal 2013, 104, 1783–1793.
[72] E. W. J. Wallace, Biophysical Journal 2013, 104, 1637–1638.
[73] T. Jetka, A. Charzyńska, A. Gambin, M. P. H. Stumpf, M. Komorowski, Bioinformatics 2014, 30, 137–138.
[74] N. Ishii, K. Nakahigashi, T. Baba, M. Robert, T. Soga, A. Kanai, T. Hirasawa, M. Naba, K. Hirai, A. Hoque, P. Y. Ho, Y. Kakazu, K. Sugawara, S. Igarashi, S. Harada, T. Masuda, N. Sugiyama, T. Togashi, M. Hasegawa, Y. Takai, K. Yugi, K. Arakawa, N. Iwata, Y. Toya, Y. Nakayama, T. Nishioka, K. Shimizu, H. Mori, M. Tomita, Science 2007, 316, 593–597.
[75] X. Y. Ni, T. Drengstig, P. Ruoff, Biophysical Journal 2009, 97, 1244–1253.
[76] H. Kitano, Nat Rev Genet 2004, 5, 826–837.
[77] A. B. Keenan, S. L. Jenkins, K. M. Jagodnik, S. Koplev, E. He, D. Torre, Z. Wang, A. B. Dohlman, M. C. Silverstein, A. Lachmann, M. V. Kuleshov, A. Ma’ayan, V. Stathias, R. Terryn, D. Cooper, M. Forlin, A. Koleti, D. Vidovic, C. Chung, S. C. Schürer, J. Vasiliauskas, M. Pilarczyk, B. Shamsaei, M. Fazel, Y. Ren, W. Niu, N. A. Clark, S. White, N. Mahi, L. Zhang, M. Kouril, J. F. Reichard, S. Sivaganesan, M. Medvedovic, J. Meller, R. J. Koch, M. R. Birtwistle, R. Iyengar, E. A. Sobie, E. U. Azeloglu, J. Kaye, J. Osterloh, K. Haston, J. Kalra, S. Finkbiener, J. Li, P. Milani, M. Adam, R. Escalante-Chong, K. Sachs, A. Lenail, D. Ramamoorthy, E. Fraenkel, G. Daigle, U. Hussain, A. Coye, J. Rothstein, D. Sareen, L. Ornelas, M. Banuelos, B. Mandefro, R. Ho, C. N. Svendsen, R. G. Lim, J. Stocksdale, M. S. Casale, T. G. Thompson, J. Wu, L. M. Thompson, V. Dardov, V. Venkatraman, A. Matlock, J. E. Van Eyk, J. D. Jaffe, M. Papanastasiou, A. Subramanian, T. R. Golub, S. D. Erickson, M. Fallahi-Sichani, M. Hafner, N. S. Gray, J.-R. Lin, C. E. Mills, J. L. Muhlich, M. Niepel, C. E. Shamu, E. H. Williams, D. Wrobel, P. K. Sorger, L. M. Heiser, J. W. Gray, J. E. Korkola, G. B. Mills, M. LaBarge, H. S. Feiler, M. A. Dane, E. Bucher, M. Nederlof, D. Sudar, S. Gross, D. F. Kilburn, R. Smith, K. Devlin, R. Margolis, L. Derr, A. Lee, A. Pillai, Cell Syst 2018, 6, 13–24.
[78] C. Feng, C. Song, Y. Liu, F. Qian, Y. Gao, Z. Ning, Q. Wang, Y. Jiang, Y. Li, M. Li, J. Chen, J. Zhang, C. Li, Nucleic Acids Research 2020, 48, D93–D100.
[79] Y. J. Cho, N. Ramakrishnan, Y. Cao, 2008, 9.
[80] C. Y. Wen, M. S. Mohamad, Advances in Bioinformatics 2011, 16.
[81] X. Zheng, Y. Huang, X. Zou, PLOS Computational Biology 2020, 16, e1007471.
[82] Z.-P. Liu, Quant Biol 2018, 6, 113–128.
[83] G. Zheng, T. Huang, in Computational Systems Biology: Methods and Protocols (Ed.: T. Huang), Springer, New York, NY, 2018, pp. 137–154.
[84] C. Ward, E. Yeung, T. Brown, B. Durtschi, S. Weyerman, R. Howes, J. Goncalves, S. Warnick, 2009, 4.
[85] D. J. Baumler, R. G. Peplinski, J. L. Reed, J. D. Glasner, N. T. Perna, BMC Syst Biol 2011, 5, 182.
[86] D. D. Payne, A. Renz, L. J. Dunphy, T. Lewis, A. Dräger, J. A. Papin, npj Syst Biol Appl 2021, 7, 37.
[87] R. Fagerberg, C. Flamm, D. Merkle, P. Peters, P. F. Stadler, Math.Comput.Sci. 2013, 7, 275–292.
[88] B. M. R. Donckels, in Encyclopedia of Systems Biology (Eds.: W. Dubitzky, O. Wolkenhauer, K.-H. Cho, H. Yokota), Springer, New York, NY, 2013, pp. 1593–1596.
[89] J. Gunawardena, BMC Biology 2014, 12, 29.
[90] U. Gupta, T. Le, W.-S. Hu, A. Bhan, P. Daoutidis, Metabolic Engineering 2018, 49, 84–93.
[91] S. Le Vent, J. Chem. Educ. 2003, 80, 89.
[92] E. O. Voit, H. A. Martens, S. W. Omholt, PLOS Computational Biology 2015, 11, e1004012.
[93] R. E. Ferner, J. K. Aronson, British Journal of Clinical Pharmacology 2016, 81, 52–55.
[94] S. Berline, C. Bricker, 1969, 46.
[95] M. A. Armstrong, Basic Topology, Springer New York, New York, NY, 1983.
[96] T. R. Maarleveld, B. G. Olivier, F. J. Bruggeman, PLOS ONE 2013, 8, e79345.
[97] F. Siso-Nadal, J. F. Ollivier, P. S. Swain, BMC Systems Biology 2007, 1, 36.
[98] A. Landeros, T. Stutz, K. Keys, A. Alekseyenko, J. S. Sinsheimer, K. Lange, M. Sehl, Comput Methods Programs Biomed 2018, 167, 23–35.
[99] F. Erhard, C. C. Friedel, R. Zimmer, BMC Bioinformatics 2008, 9, 356.
[100] P. W. Sheppard, M. Rathinam, M. Khammash, Bioinformatics 2013, 29, 140–142.
[101] O. Ostrenko, P. Incardona, R. Ramaswamy, L. Brusch, I. F. Sbalzarini, PLOS Computational Biology 2017, 13, e1005865.
[102] K. R. Sanft, S. Wu, M. Roh, J. Fu, R. K. Lim, L. R. Petzold, Bioinformatics 2011, 27, 2457–2458.
[103] A. Kazeroonian, F. Fröhlich, A. Raue, F. J. Theis, J. Hasenauer, PLOS ONE 2016, 11, e0146732.
[104] S. Hoops, S. Sahle, R. Gauges, C. Lee, J. Pahle, N. Simus, M. Singhal, L. Xu, P. Mendes, U. Kummer, Bioinformatics 2006, 22, 3067–3074.
[105] E. Kent, S. Hoops, P. Mendes, BMC Systems Biology 2012, 6, 91.
[106] M. L. Blinov, J. R. Faeder, B. Goldstein, W. S. Hlavacek, Bioinformatics 2004, 20, 3289–3291.
[107] A. Kazeroonian, F. Fröhlich, A. Raue, F. J. Theis, J. Hasenauer, PLOS ONE 2016, 11, e0146732.
[108] J. H. Abel, B. Drawert, A. Hellander, L. R. Petzold, IEEE Life Sciences Letters 2016, 2, 35–38.
[109] C. F. Lopez, J. L. Muhlich, J. A. Bachman, P. K. Sorger, Mol Syst Biol 2013, 9, 646.
[110] L. Cai, C. K. Dalal, M. B. Elowitz, Nature 2008, 455, 485–490.
[111] M. B. Elowitz, S. Leibler, Nature 2000, 403, 335–338.
[112] B. Munsky, G. Neuert, A. van Oudenaarden, Science 2012, 336, 183–187.
[113] A. Raj, S. A. Rifkin, E. Andersen, A. van Oudenaarden, Nature 2010, 463, 913–918.
[114] A. Raj, A. van Oudenaarden, Annu Rev Biophys 2009, 38, 255–270.
[115] J. M. Raser, E. K. O’Shea, Science 2005, 309, 2010–2013.
[116] E. G. Sutcliffe, A. Bartenbaker Thompson, A. R. Stettner, M. L. Marshall, M. E. Roberts, L. R. Susswein, Y. Wang, R. T. Klein, K. S. Hruska, B. D. Solomon, Fam Cancer 2019, 18, 203–209.
[117] V. Vijayan, E. K. O’Shea, J Bacteriol 2013, 195, 665–671.
[118] J. M. Raser, E. K. O’Shea, Science 2004, 304, 1811–1814.
[119] I. G. Johnston, B. Gaal, R. P. das Neves, T. Enver, F. J. Iborra, N. S. Jones, PLOS Computational Biology 2012, 8, e1002416.
[120] T. Enver, C. M. Heyworth, T. M. Dexter, Blood 1998, 92, 348–351.
[121] J. Sardanyés, T. Alarcón, Sci Rep 2018, 8, 1027.
[122] S. Adnane, A. Marino, E. Leucci, Trends in Cell Biology 2022, 32, 565–573.
[123] R. Han, G. Huang, Y. Wang, Y. Xu, Y. Hu, W. Jiang, T. Wang, T. Xiao, D. Zheng, Oncotarget 2016, 7, 72011–72020.
[124] D. Neumeier, A. Yermanos, A. Agrafiotis, L. Csepregi, T. Chowdhury, R. A. Ehling, R. Kuhn, R. B.-D. Roberto, M. Di Tacchio, R. Antonialli, D. Starkie, D. J. Lightwood, A. Oxenius, S. T. Reddy, Phenotypic Determinism and Stochasticity in Antibody Repertoires of Clonally Expanded Plasma Cells, Immunology, 2021.
[125] J. N. Carey, E. L. Mettert, M. Roggiani, K. S. Myers, P. J. Kiley, M. Goulian, Cell 2018, 173, 196-207.e14.
[126] D. Fraser, M. Kærn, Molecular Microbiology 2009, 71, 1333–1340.
[127] A. Y. Chang, W. F. Marshall, Journal of Cell Science 2017, jcs.181024.
[128] T. Stoeger, N. Battich, L. Pelkmans, Cell 2016, 164, 1151–1161.
[129] C.-C. S. Yan, C.-P. Hsu, J. Chem. Phys. 2013, 139, 224109.
[130] I. Johnston, Significance 2012, 9, 17–21.
[131] X. Sun, J. Zhang, Q. Zhao, X. Chen, W. Zhu, G. Yan, T. Zhou, BMC Syst Biol 2016, 10, 73.
[132] A. M. Bersani, E. Bersani, L. Mastroeni, Computers & Mathematics with Applications 2008, 55, 879–888.
[133] R. D. Dar, N. N. Hosmane, M. R. Arkin, R. F. Siliciano, L. S. Weinberger, Science 2014, 344, 1392–1396.
[134] D. T. Gillespie, Annu. Rev. Phys. Chem. 2007, 58, 35–55.
[135] X. Han, M. Valorani, H. N. Najm, J. Chem. Phys. 2019, 150, 194101.
[136] J. P. N. Bishwal, Parameter Estimation in Stochastic Differential Equations, Springer Berlin Heidelberg, Berlin, Heidelberg, 2008.
[137] Y. Liu, R. Gunawan, BMC Systems Biology 2014, 8, 127.
[138] E. D. Mitra, W. S. Hlavacek, Current Opinion in Systems Biology 2019, 18, 9–18.
[139] U. G. Abdulla, R. Poteau, Mathematical Biosciences 2018, 305, 133–145.
[140] N. Pullen, R. J. Morris, PLoS One 2014, 9, DOI 10.1371/journal.pone.0088419.
[141] A. P. Browning, D. J. Warne, K. Burrage, R. E. Baker, M. J. Simpson, Journal of The Royal Society Interface 2020, 17, 20200652.
[142] M. P. Saccomani, in Bioinformatics and Biomedical Engineering (Eds.: F. Ortuño, I. Rojas), Springer International Publishing, Cham, 2015, pp. 650–658.
[143] O.-T. Chis, J. R. Banga, E. Balsa-Canto, PLOS ONE 2011, 6, e27755.
[144] S. Dobre, T. Bastogne, A. Richard, IFAC Proceedings Volumes 2010, 43, 54–59.
[145] G. von Dassow, E. Meir, E. M. Munro, G. M. Odell, Nature 2000, 406, 188–192.
[146] A. Eldar, R. Dorfman, D. Weiss, H. Ashe, B.-Z. Shilo, N. Barkai, Nature 2002, 419, 304–308.
[147] E. Meir, G. von Dassow, E. Munro, G. M. Odell, Curr Biol 2002, 12, 778–786.
[148] I. Joanito, C.-C. S. Yan, J.-W. Chu, S.-H. Wu, C.-P. Hsu, PLOS Computational Biology 2020, 16, e1007740.
[149] T. Sumner, E. Shephard, I. D. L. Bogle, J R Soc Interface 2012, 9, 2156–2166.
[150] C.-P. Hsu, P.-H. Lee, C.-W. Chang, C.-T. Lee, Bioinformatics 2006, 22, 1375–1382.
[151] I. Joanito, J.-W. Chu, S.-H. Wu, C.-P. Hsu, Sci Rep 2018, 8, 13944.
[152] E. Fajiculay, C.-P. Hsu, PLOS ONE 2022, 17, e0256409.
[153] J. Paulsson, Nature 2004, 427, 415–418.
[154] H.-W. Wu, E. Fajiculay, J.-F. Wu, C.-C. S. Yan, C.-P. Hsu, S.-H. Wu, Nat. Plants 2022, 1–7.
[155] T. Okada, J.-C. Tsai, A. Mochizuki, Phys. Rev. E 2018, 98, 012417.
[156] T. Okada, A. Mochizuki, M. Furuta, J.-C. Tsai, Phys. Rev. E 2021, 103, 062212.
[157] E. Fajiculay, C.-P. Hsu, ACS Omega 2023, DOI 10.1021/acsomega.2c06113.
[158] N. Ishii, K. Nakahigashi, T. Baba, M. Robert, T. Soga, A. Kanai, T. Hirasawa, M. Naba, K. Hirai, A. Hoque, P. Y. Ho, Y. Kakazu, K. Sugawara, S. Igarashi, S. Harada, T. Masuda, N. Sugiyama, T. Togashi, M. Hasegawa, Y. Takai, K. Yugi, K. Arakawa, N. Iwata, Y. Toya, Y. Nakayama, T. Nishioka, K. Shimizu, H. Mori, M. Tomita, Science 2007, 316, 593–597.
[159] D. S. Wishart, D. Tzur, C. Knox, R. Eisner, A. C. Guo, N. Young, D. Cheng, K. Jewell, D. Arndt, S. Sawhney, C. Fung, L. Nikolai, M. Lewis, M.-A. Coutouly, I. Forsythe, P. Tang, S. Shrivastava, K. Jeroncic, P. Stothard, G. Amegbey, D. Block, David. D. Hau, J. Wagner, J. Miniaci, M. Clements, M. Gebremedhin, N. Guo, Y. Zhang, G. E. Duggan, G. D. MacInnis, A. M. Weljie, R. Dowlatabadi, F. Bamforth, D. Clive, R. Greiner, L. Li, T. Marrie, B. D. Sykes, H. J. Vogel, L. Querengesser, Nucleic Acids Res 2007, 35, D521–D526.
[160] G. Weng, U. S. Bhalla, R. Iyengar, Science 1999, 284, 92–96.
[161] J. S. Kaizer, A. K. Heller, W. L. Oberkampf, Reliability Engineering & System Safety 2015, 138, 210–218.
[162] J. R. Lumb, Mol Cell Biochem 1987, 73, 91–98.
[163] D. Richards, S. Berry, M. Howard, Cold Spring Harbor Symposia on Quantitative Biology 2012, 77, 175–181.
[164] M. R. Servedio, Y. Brandvain, S. Dhole, C. L. Fitzpatrick, E. E. Goldberg, C. A. Stern, J. V. Cleve, D. J. Yeh, PLOS Biology 2014, 12, e1002017.
[165] B. E. Mead, J. M. Karp, Genome Biology 2019, 20, 66.
[166] S. K. Palaniappan, A. Yachie-Kinoshita, S. Ghosh, in Encyclopedia of Bioinformatics and Computational Biology (Eds.: S. Ranganathan, M. Gribskov, K. Nakai, C. Schönbach), Academic Press, Oxford, 2019, pp. 789–795.
[167] G. Rozenberg, T. Bäck, J. N. Kok, Eds. , Handbook of Natural Computing, Springer-Verlag, Berlin Heidelberg, 2012.
[168] A. M. Bersani, E. Bersani, L. Mastroeni, Computers & Mathematics with Applications 2008, 55, 879–888.
[169] J. R. Faeder, M. L. Blinov, B. Goldstein, W. S. Hlavacek, Complexity 2005, 10, 22–41.
[170] N. A. W. van Riel, Briefings in Bioinformatics 2006, 7, 364–374.
[171] S.-Q. W. Yan-Yan Shen, Enz Eng 2013, 02, DOI 10.4172/2329-6674.1000113.
[172] R. Sharan, R. Shamir, Proc Int Conf Intell Syst Mol Biol 2000, 8, 307–316.
[173] C.-C. S. Yan, C.-P. Hsu, The Journal of Chemical Physics 2013, 139, 224109.
[174] R. D. Dar, N. N. Hosmane, M. R. Arkin, R. F. Siliciano, L. S. Weinberger, Science 2014, 344, 1392–1396.
[175] J. C. W. Locke, M. M. Southern, L. Kozma-Bognár, V. Hibberd, P. E. Brown, M. S. Turner, A. J. Millar, Mol Syst Biol 2005, 1, 2005.0013.
[176] X. Sun, J. Zhang, Q. Zhao, X. Chen, W. Zhu, G. Yan, T. Zhou, BMC Systems Biology 2016, 10, 73.
[177] J.-F. Wu, H.-L. Tsai, I. Joanito, Y.-C. Wu, C.-W. Chang, Y.-H. Li, Y. Wang, J. C. Hong, J.-W. Chu, C.-P. Hsu, S.-H. Wu, Nat Commun 2016, 7, 13181.
[178] J. Liu, J.-M. François, J.-P. Capp, Front. Genet. 2019, 10, DOI 10.3389/fgene.2019.00475.
[179] S. Hoops, S. Sahle, R. Gauges, C. Lee, J. Pahle, N. Simus, M. Singhal, L. Xu, P. Mendes, U. Kummer, Bioinformatics 2006, 22, 3067–3074.
[180] J. Elf, M. Ehrenberg, Genome Res 2003, 13, 2475–2484.
[181] N. G. V. Kampen, Stochastic Processes in Physics and Chemistry, North Holland, Amsterdam ; Boston, 2007.
[182] G. V. Rossum, F. L. Drake, Python 3 Reference Manual:, CreateSpace Independent Publishing Platform, Hampton, NH, 2009.
[183] S. van der Walt, S. C. Colbert, G. Varoquaux, Computing in Science Engineering 2011, 13, 22–30.
[184] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, Nature Methods 2020, 17, 261–272.
[185] A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, Am. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, Š. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, A. Scopatz, PeerJ Computer Science 2017, 3, e103.
[186] T. Okada, A. Mochizuki, Phys. Rev. Lett. 2016, 117, 048101.
[187] I. Mills, International Union of Pure and Applied Chemistry, International Union of Pure and Applied Chemistry, Eds. , Quantities, Units, and Symbols in Physical Chemistry, Blackwell Scientific Publications ; CRC Press [Distributor], Oxford ; Boston : Boca Raton, Fla, 1993.
[188] D. T. Gillespie, J. Phys. Chem. 1977, 81, 2340–2361.
[189] D. T. Gillespie, The Journal of Chemical Physics 2000, 113, 297–306.
[190] Y. Cao, D. T. Gillespie, L. R. Petzold, The Journal of Chemical Physics 2006, 124, 044109.
[191] L. Petzold, SIAM J. Sci. and Stat. Comput. 1983, 4, 136–148.
[192] U. M. Ascher, L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, 1998.
[193] W. H. Press, S. A. Teukolsky, Comput. Phys. 1992, 6, 188.
[194] D. J. Higham, 2001, 22.
[195] M. Hucka, L. Smith, F. Bergmann, S. M. Keating, 2017, DOI 10.5281/zenodo.1112521.
[196] T. W. Evans, C. S. Gillespie, D. J. Wilkinson, Bioinformatics 2008, 24, 285–286.
[197] Z. Zi, E. Klipp, Bioinformatics 2006, 22, 2704–2705.
[198] C. Cannistra, K. Medley, H. M. Sauro, SimpleSBML: A Python Package for Creating and Editing SBML Models, Systems Biology, 2015.
[199] A. Funahashi, Y. Matsuoka, A. Jouraku, M. Morohashi, N. Kikuchi, H. Kitano, Proceedings of the IEEE 2008, 96, 1254–1265.
[200] J. Hunt, Advanced Guide to Python 3 Programming, Springer International Publishing, 2019.
[201] J. Starruß, W. de Back, L. Brusch, A. Deutsch, Bioinformatics 2014, 30, 1331–1332.
[202] T. Okada, A. Mochizuki, Phys. Rev. E 2017, 96, 022322.