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Xu S, Eisenberg R, Song Z, Huang H. Coupled chemical reactions: Effects of electric field, diffusion, and boundary control. Phys Rev E 2023; 108:064413. [PMID: 38243466 DOI: 10.1103/physreve.108.064413] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2023] [Accepted: 10/31/2023] [Indexed: 01/21/2024]
Abstract
Chemical reactions involve the movement of charges, and this paper presents a mathematical model for describing chemical reactions in electrolytes. The model is developed using an energy variational method that aligns with classical thermodynamics principles. It encompasses both electrostatics and chemical reactions within consistently defined energetic and dissipative functionals. Furthermore, the energy variation method is extended to account for open systems that involve the input and output of charge and mass. Such open systems have the capability to convert one form of input energy into another form of output energy. In particular, a two-domain model is developed to study a reaction system with self-regulation and internal switching, which plays a vital role in the electron transport chain of mitochondria responsible for ATP generation-a crucial process for sustaining life. Simulations are conducted to explore the influence of electric potential on reaction rates and switching dynamics within the two-domain system. It shows that the electric potential inhibits the oxidation reaction while accelerating the reduction reaction.
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Affiliation(s)
- Shixin Xu
- Zu Chongzhi Center for Mathematics and Computational Sciences, Duke Kunshan University, 8 Duke Ave, Kunshan, Jiangsu 215316, China
| | - Robert Eisenberg
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA and Department of Physiology and Biophysics, Rush University, Chicago, Ilinois 60612, USA
| | - Zilong Song
- Math and Statistics Department, Utah State University, Old Main Hill Logan, Utah 84322, USA
| | - Huaxiong Huang
- Research Center for Mathematics, Advanced Institute of Natural Sciences, Beijing Normal University, Zhuhai, Guangdong, 519088, China; Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College, Zhuhai, Guangdong 519088, China; Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, Beijing 100875, China; and Department of Mathematics and Statistics York University, Toronto, Ontario, Canada M3J 1P3
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Eisenberg B. Setting Boundaries for Statistical Mechanics. MOLECULES (BASEL, SWITZERLAND) 2022; 27:molecules27228017. [PMID: 36432117 PMCID: PMC9696510 DOI: 10.3390/molecules27228017] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/03/2022] [Revised: 10/21/2022] [Accepted: 11/08/2022] [Indexed: 11/22/2022]
Abstract
Statistical mechanics has grown without bounds in space. Statistical mechanics of noninteracting point particles in an unbounded perfect gas is widely used to describe liquids like concentrated salt solutions of life and electrochemical technology, including batteries. Liquids are filled with interacting molecules. A perfect gas is a poor model of a liquid. Statistical mechanics without spatial bounds is impossible as well as imperfect, if molecules interact as charged particles, as nearly all atoms do. The behavior of charged particles is not defined until boundary structures and values are defined because charges are governed by Maxwell's partial differential equations. Partial differential equations require boundary structures and conditions. Boundary conditions cannot be defined uniquely 'at infinity' because the limiting process that defines 'infinity' includes such a wide variety of structures and behaviors, from elongated ellipses to circles, from light waves that never decay, to dipolar fields that decay steeply, to Coulomb fields that hardly decay at all. Boundaries and boundary conditions needed to describe matter are not prominent in classical statistical mechanics. Statistical mechanics of bounded systems is described in the EnVarA system of variational mechanics developed by Chun Liu, more than anyone else. EnVarA treatment does not yet include Maxwell equations.
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Affiliation(s)
- Bob Eisenberg
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA;
- Department of Physiology and Biophysics, Rush University Medical Center, Chicago, IL 60612, USA
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Wang H, Zou B, Su J, Wang D, Xu X. Variational methods and deep Ritz method for active elastic solids. SOFT MATTER 2022; 18:6015-6031. [PMID: 35920447 DOI: 10.1039/d2sm00404f] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
Variational methods have been widely used in soft matter physics for both static and dynamic problems. These methods are mostly based on two variational principles: the variational principle of minimum free energy (MFEVP) and Onsager's variational principle (OVP). Our interests lie in the applications of these variational methods to active matter physics. In our former work [H. Wang, T. Qian and X. Xu, Soft Matter, 2021, 17, 3634-3653], we have explored the applications of OVP-based variational methods for the modeling of active matter dynamics. In the present work, we explore variational (or energy) methods that are based on MFEVP for static problems in active elastic solids. We show that MFEVP can be used not only to derive equilibrium equations, but also to develop approximate solution methods, such as the Ritz method, for active solid statics. Moreover, the power of the Ritz-type method can be further enhanced using deep learning methods if we use deep neural networks to construct the trial functions of the variational problems. We then apply these variational methods and the deep Ritz method to study the spontaneous bending and contraction of a thin active circular plate that is induced by internal asymmetric active contraction. The circular plate is found to be bent towards its contracting side. The study of such a simple toy system gives implications for understanding the morphogenesis of solid-like confluent cell monolayers. In addition, we introduce a so-called activogravity length to characterize the importance of gravitational forces relative to internal active contraction in driving the bending of the active plate. When the lateral plate dimension is larger than the activogravity length (about 100 micron), gravitational forces become important. Such gravitaxis behaviors at multicellular scales may play significant roles in the morphogenesis and in the up-down symmetry broken during tissue development.
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Affiliation(s)
- Haiqin Wang
- Physics Program, Guangdong Technion - Israel Institute of Technology, 241 Daxue Road, Shantou, Guangdong, 515063, China.
- Technion - Israel Institute of Technology, Haifa, 32000, Israel
| | - Boyi Zou
- School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong, 518172, China
| | - Jian Su
- Physics Program, Guangdong Technion - Israel Institute of Technology, 241 Daxue Road, Shantou, Guangdong, 515063, China.
| | - Dong Wang
- School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong, 518172, China
- Shenzhen International Center for Industrial and Applied Mathematics, Shenzhen Research Institute of Big Data, Shenzhen, Guangdong, 518172, China
| | - Xinpeng Xu
- Physics Program, Guangdong Technion - Israel Institute of Technology, 241 Daxue Road, Shantou, Guangdong, 515063, China.
- Technion - Israel Institute of Technology, Haifa, 32000, Israel
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Wang Y, Liu C. Some Recent Advances in Energetic Variational Approaches. ENTROPY 2022; 24:e24050721. [PMID: 35626604 PMCID: PMC9141973 DOI: 10.3390/e24050721] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/16/2022] [Revised: 04/04/2022] [Accepted: 05/06/2022] [Indexed: 11/16/2022]
Abstract
In this paper, we summarize some recent advances related to the energetic variational approach (EnVarA), a general variational framework of building thermodynamically consistent models for complex fluids, by some examples. Particular focus will be placed on how to model systems involving chemo-mechanical couplings and non-isothermal effects.
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Budithi A, Su S, Kirshtein A, Shahriyari L. Data Driven Mathematical Model of FOLFIRI Treatment for Colon Cancer. Cancers (Basel) 2021; 13:2632. [PMID: 34071939 PMCID: PMC8198096 DOI: 10.3390/cancers13112632] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2021] [Revised: 05/21/2021] [Accepted: 05/21/2021] [Indexed: 12/12/2022] Open
Abstract
Many colon cancer patients show resistance to their treatments. Therefore, it is important to consider unique characteristic of each tumor to find the best treatment options for each patient. In this study, we develop a data driven mathematical model for interaction between the tumor microenvironment and FOLFIRI drug agents in colon cancer. Patients are divided into five distinct clusters based on their estimated immune cell fractions obtained from their primary tumors' gene expression data. We then analyze the effects of drugs on cancer cells and immune cells in each group, and we observe different responses to the FOLFIRI drugs between patients in different immune groups. For instance, patients in cluster 3 with the highest T-reg/T-helper ratio respond better to the FOLFIRI treatment, while patients in cluster 2 with the lowest T-reg/T-helper ratio resist the treatment. Moreover, we use ROC curve to validate the model using the tumor status of the patients at their follow up, and the model predicts well for the earlier follow up days.
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Affiliation(s)
- Aparajita Budithi
- Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA; (A.B.); (S.S.)
| | - Sumeyye Su
- Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA; (A.B.); (S.S.)
| | - Arkadz Kirshtein
- Department of Mathematics, Tufts University, Medford, MA 02155, USA;
| | - Leili Shahriyari
- Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA; (A.B.); (S.S.)
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