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Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics

Many emergent, non-fundamental models of complex systems can be described naturally by the temporal evolution of spatial structures with some nontrivial discretized topology, such as a graph with suitable parameter vectors labeling its vertices. For example, the cytoskeleton of a single cell, such a...

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Autor principal: Mjolsness, Eric
Formato: Online Artículo Texto
Lenguaje:English
Publicado: 2022
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9879069/
https://www.ncbi.nlm.nih.gov/pubmed/36712785
http://dx.doi.org/10.3389/fsysb.2022.898858
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author Mjolsness, Eric
author_facet Mjolsness, Eric
author_sort Mjolsness, Eric
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description Many emergent, non-fundamental models of complex systems can be described naturally by the temporal evolution of spatial structures with some nontrivial discretized topology, such as a graph with suitable parameter vectors labeling its vertices. For example, the cytoskeleton of a single cell, such as the cortical microtubule network in a plant cell or the actin filaments in a synapse, comprises many interconnected polymers whose topology is naturally graph-like and dynamic. The same can be said for cells connected dynamically in a developing tissue. There is a mathematical framework suitable for expressing such emergent dynamics, “stochastic parameterized graph grammars,” composed of a collection of the graph- and parameter-altering rules, each of which has a time-evolution operator that suitably moves probability. These rule-level operators form an operator algebra, much like particle creation/annihilation operators or Lie group generators. Here, we present an explicit and constructive calculation, in terms of elementary basis operators and standard component notation, of what turns out to be a general combinatorial expression for the operator algebra that reduces products and, therefore, commutators of graph grammar rule operators to equivalent integer-weighted sums of such operators. We show how these results extend to “dynamical graph grammars,” which include rules that bear local differential equation dynamics for some continuous-valued parameters. Commutators of such time-evolution operators have analytic uses, including deriving efficient simulation algorithms and approximations and estimating their errors. The resulting formalism is complementary to spatial models in the form of partial differential equations or stochastic reaction-diffusion processes. We discuss the potential application of this framework to the remodeling dynamics of the microtubule cytoskeleton in cortical microtubule networks relevant to plant development and of the actin cytoskeleton in, for example, a growing or shrinking synaptic spine head. Both cytoskeletal systems underlie biological morphodynamics.
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spelling pubmed-98790692023-01-26 Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics Mjolsness, Eric Front Syst Biol Article Many emergent, non-fundamental models of complex systems can be described naturally by the temporal evolution of spatial structures with some nontrivial discretized topology, such as a graph with suitable parameter vectors labeling its vertices. For example, the cytoskeleton of a single cell, such as the cortical microtubule network in a plant cell or the actin filaments in a synapse, comprises many interconnected polymers whose topology is naturally graph-like and dynamic. The same can be said for cells connected dynamically in a developing tissue. There is a mathematical framework suitable for expressing such emergent dynamics, “stochastic parameterized graph grammars,” composed of a collection of the graph- and parameter-altering rules, each of which has a time-evolution operator that suitably moves probability. These rule-level operators form an operator algebra, much like particle creation/annihilation operators or Lie group generators. Here, we present an explicit and constructive calculation, in terms of elementary basis operators and standard component notation, of what turns out to be a general combinatorial expression for the operator algebra that reduces products and, therefore, commutators of graph grammar rule operators to equivalent integer-weighted sums of such operators. We show how these results extend to “dynamical graph grammars,” which include rules that bear local differential equation dynamics for some continuous-valued parameters. Commutators of such time-evolution operators have analytic uses, including deriving efficient simulation algorithms and approximations and estimating their errors. The resulting formalism is complementary to spatial models in the form of partial differential equations or stochastic reaction-diffusion processes. We discuss the potential application of this framework to the remodeling dynamics of the microtubule cytoskeleton in cortical microtubule networks relevant to plant development and of the actin cytoskeleton in, for example, a growing or shrinking synaptic spine head. Both cytoskeletal systems underlie biological morphodynamics. 2022-09 2022-09-09 /pmc/articles/PMC9879069/ /pubmed/36712785 http://dx.doi.org/10.3389/fsysb.2022.898858 Text en https://creativecommons.org/licenses/by/4.0/This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
spellingShingle Article
Mjolsness, Eric
Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics
title Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics
title_full Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics
title_fullStr Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics
title_full_unstemmed Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics
title_short Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics
title_sort explicit calculation of structural commutation relations for stochastic and dynamical graph grammar rule operators in biological morphodynamics
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9879069/
https://www.ncbi.nlm.nih.gov/pubmed/36712785
http://dx.doi.org/10.3389/fsysb.2022.898858
work_keys_str_mv AT mjolsnesseric explicitcalculationofstructuralcommutationrelationsforstochasticanddynamicalgraphgrammarruleoperatorsinbiologicalmorphodynamics