Patterning by genetic networks and modular principle

S. Genieys; S. Vakulenko

Home/Archive/arXiv

arXiv · math/0502089

Patterning by genetic networks and modular principle

S. Genieys·S. Vakulenko

View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider here the morphogenesis (pattern formation) problem for some genetic network models. First, we show that any given spatio-temporal pattern can be generated by a genetic network involving a sufficiently large number of genes. Moreover, patterning process can be performed by an effective algorithm. We also show that Turing's or Meinhardt's type reaction-diffusion models can be approximated by genetic networks. These results exploit the fundamental fact that the genes form functional units and are organised in blocks (modular principle). Due to this modular organisation, the genes always are capable to construct any new patterns and even any time sequences of new patterns from old patterns. Computer simulations illustrate analytical results.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

S. Genieys, S. Vakulenko. 2005-02-04. Patterning by genetic networks and modular principle. https://arxiv.org/abs/math/0502089

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Metastability Landscapes in Nonlinear Ghost-Normal Scalar Fields

We investigate finite time dynamics in nonlinear classical scalar field systems containing one ghost and one normal kinetic sector. Using a spacetime finite element formulation, we study the dependence of threshold crossing and numerical continuation times on initial amplitude, characteristic wavenumber, spectral tilt, phase structure, and spatial localization. In the quartic model, larger amplitudes generally lead to earlier growth, while higher wavenumbers and ultraviolet-biased spectra favor longer resolved bounded evolution. Smooth compact initial data exhibit a broad region with no observed growth threshold crossing over the simulated interval. A direct comparison of two Gaussian evolutions shows that enhanced low-wavenumber power can precede rapid field growth, although this observation does not establish a universal spectral-transfer mechanism. We also compare the quartic interaction with a lifted $ϕ^6$ potential. The latter gives a markedly different continuation landscape, with a broad long-converged region followed by shorter accessible terminal times at larger amplitudes. Localization diagnostics do not identify a persistent oscillon core in the selected cases. By distinguishing amplitude growth, spectral reorganization, solver-limited termination, and finite-time censoring, our results characterize a nontrivial metastability landscape without implying global or quantum stability.

math.DS↗

arXiv2026

Slow Entropy of Quasi-Unipotent Actions on $S$-Arithmetic Homogeneous Spaces

We study metric and topological slow entropy for translations on a class of finite-volume $S$-arithmetic homogeneous spaces with real and non-Archimedean factors. We consider translations whose real component is quasi-unipotent and whose non-Archimedean component lies in a compact open subgroup. At polynomial scales, we show that the metric and topological slow entropies coincide and are determined by the chain structure of the real adjoint action. In particular, the non-Archimedean component contributes no additional polynomial slow entropy. Our result extends the slow entropy formula of Kanigowski, Vinhage, and Wei for quasi-unipotent flows on real homogeneous spaces to this $S$-arithmetic setting.

math.DS↗

arXiv2026

Discrete history operators: Exact state-space realization for sequence memory

A prescribed finite-dimensional analysis of an evolving discrete history need not admit an exact recurrence driven only by its current state and the entering input. We study exact realizations of causal linear discrete history operators (DHOs). Operator factorization and kernel inclusion characterize exact closure, while the cross-projection defect characterizes closure failure for orthonormal analyses. For finite-dimensional history spaces over fixed finite horizons, we distinguish two minimality problems. Reachable task quotients determine the minimum state dimension for preserving prescribed future tasks on reachable histories. Backward invariant envelopes characterize the minimum dimension of exact linear realizations recovering prescribed analyses on full history spaces within the admissible class. The minimum number of additional independent state directions is given by the rank of the propagated closure defect operator restricted to the kernel of the current analysis. For geometrically weighted countable histories, matrix mode transport yields structured exact recurrences and a uniform exponential decay estimate for state transitions. Additional conditions provide uniform Gram bounds and refined decay estimates. A stationary Meixner specialization yields explicit recurrence matrices, an O(N) arithmetic state update, and minimum augmentation dimensions over finite horizons for selected coefficients. Numerical experiments examine recurrence accuracy, state transition bounds, Gram conditions, and minimum realization dimensions.

math.DS↗