An Accurate and Stable Filtered Explicit Scheme for Biopolymerization Processes in the Presence of Perturbations

Lisa Davis; Faranak Pahlevani; Timmy Susai Rajan

doi:doi:10.11648/j.acm.20211006.11

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An Accurate and Stable Filtered Explicit Scheme for Biopolymerization Processes in the Presence of Perturbations

Lisa Davis, Faranak Pahlevani, Timmy Susai Rajan

Published in Applied and Computational Mathematics (Volume 10, Issue 6)

Received: 31 August 2021 Accepted: 13 October 2021 Published: 5 November 2021

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Abstract

The focus of this paper is the development, numerical simulation and parameter analysis of a model of the transcription of ribosomal RNA in highly transcribed genes. Inspired by the well-known classic Lighthill-Whitham-Richards (LWR) traffic flow model, a linear advection continuum model is used to describe the DNA transcription process. In this model, elongation velocity is assumed to be essentially constant as RNA polymerases move along the strand through different phases of gene transcription. One advantage of using the linear model is that it allows one to quantify how small perturbations in elongation velocity and inflow parameters affect important biology measures such as Average Transcription Time (ATT) for the gene. The ATT per polymerase is the amount of time an individual RNAP spends traveling through the DNA strand. The numerical treatment for model simulations includes introducing a low complexity and time accurate method by adding a simple linear time filter to the classic upwind scheme. This improved method is modular and requires a minimal modification of adding only one line of code resulting in increased accuracy without increased computational expense. In addition, it removes the overdamping of upwind. A stability condition for the new algorithm is derived, and numerical computations illustrate stability and convergence of the filtered scheme as well as improved ATT estimation.

Published in	Applied and Computational Mathematics (Volume 10, Issue 6)
DOI	10.11648/j.acm.20211006.11
Page(s)	121-137
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Advection Equation, Lighthill-Whitham-Richards Model, Ribosomal RNA, RNA Polymerases, Time Filter, Traffic Flow Models, Transcription Time, Upwind Scheme

References

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Lisa Davis, Faranak Pahlevani, Timmy Susai Rajan. (2021). An Accurate and Stable Filtered Explicit Scheme for Biopolymerization Processes in the Presence of Perturbations. Applied and Computational Mathematics, 10(6), 121-137. https://doi.org/10.11648/j.acm.20211006.11

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ACS Style

Lisa Davis; Faranak Pahlevani; Timmy Susai Rajan. An Accurate and Stable Filtered Explicit Scheme for Biopolymerization Processes in the Presence of Perturbations. Appl. Comput. Math. 2021, 10(6), 121-137. doi: 10.11648/j.acm.20211006.11

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AMA Style

Lisa Davis, Faranak Pahlevani, Timmy Susai Rajan. An Accurate and Stable Filtered Explicit Scheme for Biopolymerization Processes in the Presence of Perturbations. Appl Comput Math. 2021;10(6):121-137. doi: 10.11648/j.acm.20211006.11

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@article{10.11648/j.acm.20211006.11,
  author = {Lisa Davis and Faranak Pahlevani and Timmy Susai Rajan},
  title = {An Accurate and Stable Filtered Explicit Scheme for Biopolymerization Processes in the Presence of Perturbations},
  journal = {Applied and Computational Mathematics},
  volume = {10},
  number = {6},
  pages = {121-137},
  doi = {10.11648/j.acm.20211006.11},
  url = {https://doi.org/10.11648/j.acm.20211006.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211006.11},
  abstract = {The focus of this paper is the development, numerical simulation and parameter analysis of a model of the transcription of ribosomal RNA in highly transcribed genes. Inspired by the well-known classic Lighthill-Whitham-Richards (LWR) traffic flow model, a linear advection continuum model is used to describe the DNA transcription process. In this model, elongation velocity is assumed to be essentially constant as RNA polymerases move along the strand through different phases of gene transcription. One advantage of using the linear model is that it allows one to quantify how small perturbations in elongation velocity and inflow parameters affect important biology measures such as Average Transcription Time (ATT) for the gene. The ATT per polymerase is the amount of time an individual RNAP spends traveling through the DNA strand. The numerical treatment for model simulations includes introducing a low complexity and time accurate method by adding a simple linear time filter to the classic upwind scheme. This improved method is modular and requires a minimal modification of adding only one line of code resulting in increased accuracy without increased computational expense. In addition, it removes the overdamping of upwind. A stability condition for the new algorithm is derived, and numerical computations illustrate stability and convergence of the filtered scheme as well as improved ATT estimation.},
 year = {2021}
}

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TY  - JOUR
T1  - An Accurate and Stable Filtered Explicit Scheme for Biopolymerization Processes in the Presence of Perturbations
AU  - Lisa Davis
AU  - Faranak Pahlevani
AU  - Timmy Susai Rajan
Y1  - 2021/11/05
PY  - 2021
N1  - https://doi.org/10.11648/j.acm.20211006.11
DO  - 10.11648/j.acm.20211006.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
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PB  - Science Publishing Group
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AB  - The focus of this paper is the development, numerical simulation and parameter analysis of a model of the transcription of ribosomal RNA in highly transcribed genes. Inspired by the well-known classic Lighthill-Whitham-Richards (LWR) traffic flow model, a linear advection continuum model is used to describe the DNA transcription process. In this model, elongation velocity is assumed to be essentially constant as RNA polymerases move along the strand through different phases of gene transcription. One advantage of using the linear model is that it allows one to quantify how small perturbations in elongation velocity and inflow parameters affect important biology measures such as Average Transcription Time (ATT) for the gene. The ATT per polymerase is the amount of time an individual RNAP spends traveling through the DNA strand. The numerical treatment for model simulations includes introducing a low complexity and time accurate method by adding a simple linear time filter to the classic upwind scheme. This improved method is modular and requires a minimal modification of adding only one line of code resulting in increased accuracy without increased computational expense. In addition, it removes the overdamping of upwind. A stability condition for the new algorithm is derived, and numerical computations illustrate stability and convergence of the filtered scheme as well as improved ATT estimation.
VL  - 10
IS  - 6
ER  -

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Author Information

Lisa Davis

Department of Mathematical Sciences, Montana State University, Bozeman, United States
Faranak Pahlevani

Division of Science & Engineering, Penn State University-Abington, Abington, United States
Timmy Susai Rajan

Division of Science & Engineering, Penn State University-Abington, Abington, United States

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APA Style

Lisa Davis, Faranak Pahlevani, Timmy Susai Rajan. (2021). An Accurate and Stable Filtered Explicit Scheme for Biopolymerization Processes in the Presence of Perturbations. Applied and Computational Mathematics, 10(6), 121-137. https://doi.org/10.11648/j.acm.20211006.11

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ACS Style

Lisa Davis; Faranak Pahlevani; Timmy Susai Rajan. An Accurate and Stable Filtered Explicit Scheme for Biopolymerization Processes in the Presence of Perturbations. Appl. Comput. Math. 2021, 10(6), 121-137. doi: 10.11648/j.acm.20211006.11

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AMA Style

Lisa Davis, Faranak Pahlevani, Timmy Susai Rajan. An Accurate and Stable Filtered Explicit Scheme for Biopolymerization Processes in the Presence of Perturbations. Appl Comput Math. 2021;10(6):121-137. doi: 10.11648/j.acm.20211006.11

Copy | Download

@article{10.11648/j.acm.20211006.11,
  author = {Lisa Davis and Faranak Pahlevani and Timmy Susai Rajan},
  title = {An Accurate and Stable Filtered Explicit Scheme for Biopolymerization Processes in the Presence of Perturbations},
  journal = {Applied and Computational Mathematics},
  volume = {10},
  number = {6},
  pages = {121-137},
  doi = {10.11648/j.acm.20211006.11},
  url = {https://doi.org/10.11648/j.acm.20211006.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211006.11},
  abstract = {The focus of this paper is the development, numerical simulation and parameter analysis of a model of the transcription of ribosomal RNA in highly transcribed genes. Inspired by the well-known classic Lighthill-Whitham-Richards (LWR) traffic flow model, a linear advection continuum model is used to describe the DNA transcription process. In this model, elongation velocity is assumed to be essentially constant as RNA polymerases move along the strand through different phases of gene transcription. One advantage of using the linear model is that it allows one to quantify how small perturbations in elongation velocity and inflow parameters affect important biology measures such as Average Transcription Time (ATT) for the gene. The ATT per polymerase is the amount of time an individual RNAP spends traveling through the DNA strand. The numerical treatment for model simulations includes introducing a low complexity and time accurate method by adding a simple linear time filter to the classic upwind scheme. This improved method is modular and requires a minimal modification of adding only one line of code resulting in increased accuracy without increased computational expense. In addition, it removes the overdamping of upwind. A stability condition for the new algorithm is derived, and numerical computations illustrate stability and convergence of the filtered scheme as well as improved ATT estimation.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - An Accurate and Stable Filtered Explicit Scheme for Biopolymerization Processes in the Presence of Perturbations
AU  - Lisa Davis
AU  - Faranak Pahlevani
AU  - Timmy Susai Rajan
Y1  - 2021/11/05
PY  - 2021
N1  - https://doi.org/10.11648/j.acm.20211006.11
DO  - 10.11648/j.acm.20211006.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 121
EP  - 137
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20211006.11
AB  - The focus of this paper is the development, numerical simulation and parameter analysis of a model of the transcription of ribosomal RNA in highly transcribed genes. Inspired by the well-known classic Lighthill-Whitham-Richards (LWR) traffic flow model, a linear advection continuum model is used to describe the DNA transcription process. In this model, elongation velocity is assumed to be essentially constant as RNA polymerases move along the strand through different phases of gene transcription. One advantage of using the linear model is that it allows one to quantify how small perturbations in elongation velocity and inflow parameters affect important biology measures such as Average Transcription Time (ATT) for the gene. The ATT per polymerase is the amount of time an individual RNAP spends traveling through the DNA strand. The numerical treatment for model simulations includes introducing a low complexity and time accurate method by adding a simple linear time filter to the classic upwind scheme. This improved method is modular and requires a minimal modification of adding only one line of code resulting in increased accuracy without increased computational expense. In addition, it removes the overdamping of upwind. A stability condition for the new algorithm is derived, and numerical computations illustrate stability and convergence of the filtered scheme as well as improved ATT estimation.
VL  - 10
IS  - 6
ER  -

Copy | Download