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Molecular Synthesis for Probability Theory and Stochastic Process

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Abstract

It is common that probability theory and stochastic process, especially Markov chains, have long been used to study and explain the behaviors of chemical reaction networks (CRNs). Nonetheless, this paper sees things from a reverse angle, devoting itself in synthesizing common probability theory and stochastic process with CRNs. The main motivation is to imitate and explore the evolution of large-scale and complex practical systems based on CRNs, by making use of the inherent parallelism and randomness. In our conference paper, a preliminary but concise approach has been put forward to synthesize the stand-alone examples such as law of total probability, Bayes’ theorem, and n-step transition of Markov chains. To make this methodology systematic and theoretically sound, we enrich and offer more solid foundation for the previous version. Rigorous stability analysis based on ordinary differential equations (ODEs) are provided. This paper further deeply discusses and distinguishes building stochastic models for CRNs and utilizing CRNs to solve stochastic problems. A joint distribution of Markov chains are implemented using molecular reactions as a showcase. In order to enhance the clearness of the results, all the simulations are done according to deterministic mass action. It is worth noting that an already mathematically proven conclusion, which states that nearly an arbitrary set of bimolecular or unimolecular reactions can be implemented by DNA strand displacement reactions, ensures the meaningfulness of our work. It is also believed that, though in its infancy, the proposed approach is also valid for other molecular synthesis than DNA, as long as the kinetics constraints are met.

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Acknowledgements

The authors thank Dr. Xiaoqing Chen for her consistent support to our project. This work is partially supported by NSFC under grants 61501116 and 61571105, Jiangsu Provincial NSF under grant BK20140636, State Key Laboratory of ASIC & System under grant 2016KF007, Student Research Training Program of SEU, Huawei HIRP Flagship under grant YB201504, Intel Collaborative Research Institute for MNC, the Fundamental Research Funds for the Central Universities, and the Project Sponsored by the SRF for the Returned Overseas Chinese Scholars of State Education Ministry.

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Authors and Affiliations

  1. Laboratory of Efficient Architectures for Digital-communication and Signal-processing (LEADS), National Mobile Communications Research Laboratory, Southeast University, Nanjing, China

    Ziyuan Shen, Lulu Ge, Zaichen Zhang, Xiaohu You & Chuan Zhang

  2. State Key Laboratory of Coordination Chemistry, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing, China

    Wei Wei & Jing Zhao

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  1. Ziyuan Shen
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  2. Lulu Ge
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  3. Wei Wei
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  4. Jing Zhao
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  5. Zaichen Zhang
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  6. Xiaohu You
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  7. Chuan Zhang
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Correspondence to Chuan Zhang.

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Shen, Z., Ge, L., Wei, W. et al. Molecular Synthesis for Probability Theory and Stochastic Process. J Sign Process Syst 90, 1479–1494 (2018). https://doi.org/10.1007/s11265-017-1318-7

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  • DOI: https://doi.org/10.1007/s11265-017-1318-7

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