[Paper Review] Asymptotic state lumping in transport and diffusion problems on networks

Jacek Banasiak, Aleksandra Falkiewicz+1 more|arXiv (Cornell University)|Mar 2, 2015

Spectral Theory in Mathematical Physics34 references3 citations

TL;DR

This paper proposes a rigorous asymptotic framework for aggregating detailed microscale transport and diffusion models on networks into a simplified macroscale model, demonstrating that fast internal dynamics (diffusion/transport) on edges combined with slow inter-node exchange leads to solutions that converge to a reduced system governed by ordinary differential equations. The key contribution is proving that the aggregated dynamics asymptotically matches the original macroscale model, with the network structure preserved through stable distribution vectors in the limit.

ABSTRACT

One of the aims of systems biology is to build multiple layered and multiple scale models of living systems which can efficiently describe phenomena occurring at various level of resolution. Such models should consist of layers of various microsystems interconnected by a network of pathways, to form a macrosystem in a consistent way; that is, the observable characteristics of the macrosystem should be, at least asymptotically, derivable by aggregation of the appropriate features of the microsystems forming it, and from the properties of the network. In this paper we consider a general macromodel describing a population consisting of several interacting with each other subgroups, with the rules of interactions given by a system of ordinary differential equations, and we construct two different micromodels whose aggregated dynamics is approximately the same as that of the original macromodel. The micromodels offer a more detailed description of the original macromodel's dynamics by considering an internal structure of each subgroup. Here, each subgroup is represented by an edge of a graph with diffusion or transport occurring along it, while the interactions between the edges are described by interface conditions at the nodes joining them. We prove that with an appropriate scaling of such models, roughly speaking, with fast diffusion or transport combined with slow exchange at the nodes, the solutions of the micromodels are close to the solution to the macromodel.

Motivation & Objective

• To develop a mathematical framework for aggregating detailed microscale models on networks into a simplified macroscale model.
• To address the challenge of multiscale dynamics in systems biology where different processes occur at vastly different time scales.
• To show that fast internal dynamics on network edges lead to effective aggregation of states, preserving network structure in the limit.
• To establish conditions under which micromodels with internal structure converge to a macroscale ODE system.
• To formalize the concept of 'asymptotic state lumping' in the context of transport and diffusion on graphs.

Proposed method

• Model the macroscale system as a system of ODEs describing population dynamics across interconnected subgroups.
• Construct micromodels where each subgroup is represented as an edge in a graph, with transport or diffusion dynamics governed by PDEs on each edge.
• Introduce a small parameter ǫ to scale the rate of inter-node exchange, assuming it is much slower than internal dynamics on edges.
• Use semigroup theory and spectral analysis to study the long-time behavior of the micromodels.
• Apply asymptotic analysis to show that the projection of the micromodel solution onto the slow manifold converges to the macroscale ODE solution.
• Leverage Perron-Frobenius theory to characterize the stable distribution vector N, which encodes the asymptotic fraction of mass on each edge.

Experimental results

Research questions

• RQ1 Under what conditions does a micromodel with internal structure on a network converge to a simplified macroscale ODE model?
• RQ2 How is the network topology preserved in the asymptotic limit despite aggregation of internal states?
• RQ3 What role does the Perron eigenvector of the transition matrix play in the asymptotic aggregation process?
• RQ4 Can transport and diffusion processes on edges be effectively lumped into a single macroscale ODE system when inter-node exchange is slow?
• RQ5 How does the initial layer behavior affect the convergence of micromodel solutions to the macroscale solution?

Key findings

• The projection of the micromodel solution onto the slow manifold converges to the solution of the macroscale ODE system as ǫ → 0.
• The asymptotic distribution of mass across the network is determined by the Perron eigenvector N of the transition matrix K, which encodes the stable patch distribution.
• The aggregated macroscale model (85) retains the influence of the original network structure through the coefficients µ∗ and β∗, which are weighted averages of patch-specific rates.
• The initial layer behavior in the micromodel does not decay exponentially, indicating that transient dynamics are non-trivial and must be accounted for in the asymptotic analysis.
• The micromodels constructed via transport or diffusion on edges yield the same asymptotic macroscale dynamics as the original ODE system, validating the lumping procedure.
• The convergence is established via semigroup theory and spectral analysis, showing that the slow dynamics are governed by the kernel of the projection operator P.