Time and space generalized diffusion equation on graphs/networks

TL;DR

This paper introduces a generalized diffusion equation on graphs using fractional derivatives and transformed Laplacians, analytically solving it and demonstrating its ability to model various diffusion regimes, including complex alternations.

Contribution

It develops a unified framework for normal and anomalous diffusion on networks, including an extension to model alternating diffusion regimes relevant for biological systems.

Findings

Analytical solutions cover normal, sub-, and superdiffusion regimes.

Alternating diffusion regimes enhance exploration efficiency in DNA-like systems.

Simulation shows benefits of diffusion alternation for protein-DNA interactions.

Abstract

Normal and anomalous diffusion are ubiquitous in many complex systems [1] . Here, we define a time and space generalized diffusion equation (GDE), which uses fractional-time derivatives and transformed d-path Laplacian operators on graphs/networks. We find analytically the solution of this equation and prove that it covers the regimes of normal, sub- and superdiffusion as a function of the two parameters of the model. We extend the GDE to consider a system with temporal alternancy of normal and anomalous diffusion which can be observed for instance in the diffusion of proteins along a DNA chain. We perform computational experiments on a one-dimensional system emulating a linear DNA chain. It is shown that a subdiffusive-superdiffusive alternant regime allows the diffusive particle to explore more slowly small regions of the chain with a faster global exploration, than a subdiffusive-subdiffusive regime. Therefore, an alternancy of sliding (subdiffusive) with hopping and intersegmental transfer (superdiffusive) mechanisms show important advances for protein-DNA interactions.