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

Abstract

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.

Approximate simulation of cortical microtubule models using dynamical graph grammars

“…Equation (8b) uses the fact that the resulting cells of fixed dimension are all well-separated geometrically with enough margin (due to the ‘collar’ of dimension [ 21 ]) so that reaction instances commute to high accuracy if they are assigned to different cells of the same dimensionality, by some reaction instance allocation function . The commutators of equation (8b) can be calculated as derived in [ 31 ], but they will inherit the exponential falloff with separation that we assumed for the rule propensities (see [ 31 ], equation 12 therein). Hence, the dynamics of different cells of the same original dimension and can be simulated in any order, or in parallel, at little cost in accuracy.…”

“…It can be seen that without domain subdivisions, the approximate algorithm reduces to the exact algorithm. A more complete mathematical treatment of the approximate algorithm using DGG commutators computed as in [ 31 ] to bound the operator splitting errors will be the topic of future work.…”