Reversibility and irreversibility in stochastic chemical kinetics

Малышев, В. А.; Pirogov, S A

doi:10.1070/rm2008v063n01abeh004500

Cited by 18 publications

(13 citation statements)

References 34 publications

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“…It also differs from those focusing on moments (Gadgil et al, 2006) but leans towards those of nonequilibrium thermodynamics approaches (Qian, 2005). In addition, it differs from other more formal stochastic approaches in biology but more from a mathematical point of view (Malyshev and Pirogov, 2008). The powerful adaptive landscape concept was first proposed by a great biologist a long time ago (Wright, 1932).…”

Section: Introductionmentioning

confidence: 99%

Global view of bionetwork dynamics: adaptive landscape

2009

Journal of Genetics and Genomics

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Based on recent work, I will give a nontechnical brief review of a powerful quantitative concept in biology, adaptive landscape, initially proposed by S. Wright over 70 years ago, reintroduced by one of the founders of molecular biology and by others in different biological contexts, but apparently forgotten by modern biologists for many years. Nevertheless, this concept finds an increasingly important role in the development of systems biology and bionetwork dynamics modeling, from phage lambda genetic switch to endogenous network for cancer genesis and progression. It is an ideal quantification to describe the robustness and stability of bionetworks. Here, I will first introduce five landmark proposals in biology on this concept, to demonstrate an important common thread in theoretical biology. Then I will discuss a few recent results, focusing on the studies showing theoretical consistency of adaptive landscape. From the perspective of a working scientist and of what is needed logically for a dynamical theory when confronting empirical data, the adaptive landscape is useful both metaphorically and quantitatively, and has captured an essential aspect of biological dynamical processes. Though at the theoretical level the adaptive landscape must exist and it can be used across hierarchical boundaries in biology, many associated issues are indeed vague in their initial formulations and their quantitative realizations are not easy, and are good research topics for quantitative biologists. I will discuss three types of open problems associated with the adaptive landscape in a broader perspective.

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Section: Introductionmentioning

confidence: 99%

Global view of bionetwork dynamics: adaptive landscape

2009

Journal of Genetics and Genomics

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“…The equations for complex physicochemical processes in the general form are written as [6,7,11,12,24,25]:…”

Section: The Methods and Definitionsmentioning

confidence: 99%

“…The first of them formulates the conservation law of the number of atoms (or complexes of atoms) of the form A: N 1 +N 3 , the second one is the number of atoms of the form B, the second one is the number of atoms of the form B: N 2 + N 3 , and the third law is the number of atoms of the form C: N 2 + N 4 . It is easy to verify that a linear operator: I µ (t) = n i=1 µ i N i (t) = (µ, N) is conserved along the solutions of the system under consideration then and only then when the vector µ is orthogonal to all Boltzmann-Orlov-Moser-Bruno vectors [12,[24][25][26][27] α − β. In this example, the Boltzmann-Orlov-Moser-Bruno vector is unique up to a factor: α − β = (1, 1, −1, −1).…”

Section: The Methods and Definitionsmentioning

confidence: 99%

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Kinetic Equations for Particle Clusters Differing in Shape and the H-theorem

Adzhiev

Batishcheva

Мелихов

et al. 2019

Physics

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The question of constructing models for the evolution of clusters that differ in shape based on the Boltzmann’s H-theorem is investigated. The first, simplest kinetic equations are proposed and their properties are studied: the conditions for fulfilling the H-theorem (the conditions for detailed and semidetailed balance). These equations are to generalize the classical coagulation–fragmentation type equations for cases when not only mass but also particle shape is taken into account. To construct correct (physically grounded) kinetic models, the fulfillment of the condition of detailed balance is shown to be necessary to monitor, since it is proved that for accepted frequency functions, the condition of detailed balance is fulfilled and the H-theorem is valid. It is shown that for particular and very important cases, the H-theorem holds: the fulfillment of the Arrhenius law and the additivity of the activation energy for interacting particles are found to be essential. In addition, based on the connection of the principle of detailed balance with the Boltzmann equation for the probability of state, the expressions for the reaction rate coefficients are obtained.

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“…4. Проверить, что во всех этих случаях пределы временных средних совпадают с экстремалями Больцмана, обобщая и упрощая результаты работ [1]- [28].…”

Section: Introductionunclassified

Энтропия По Больцману И Пуанкаре

Веденяпин¹,

Vedenyapin²,

Аджиев³

et al. 2014

Uspekhi Mat. Nauk

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В работе доказывается H-теорема для обобщений уравнений химической кинетики. Рассматриваются важные физические примеры такого обобщения: дискретные модели квантовых кинетических уравнений (уравнений Юлинга-Уленбека) и квантовый марковский процесс (квантовое случайное блуждание). Доказывается совпадение временных средних с экстремалями по Больцману для всех таких уравнений, а также для уравнения Лиувилля. Библиография: 41 название. Ключевые слова: уравнение Больцмана, H-теорема, энтропия, законы сохранения, дискретная модель, экстремаль по Больцману, уравнение Лиувилля, временное среднее, среднее по Чезаро (Cesaro), чезаровское среднее, марковские цепи, вариационный принцип.

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Reversibility and irreversibility in stochastic chemical kinetics

Cited by 18 publications

References 34 publications

Global view of bionetwork dynamics: adaptive landscape

Global view of bionetwork dynamics: adaptive landscape

Kinetic Equations for Particle Clusters Differing in Shape and the H-theorem

Энтропия По Больцману И Пуанкаре

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