Continuum Modeling of Biological Network Formation

Albi, Giacomo; Burger, Martin; Haškovec, Jan; Markowich, Peter A.; Schlottbom, Matthias

doi:10.1007/978-3-319-49996-3_1

Cited by 26 publications

(56 citation statements)

References 42 publications

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“…Given the vector of conductivities C = (C ij ) (i,j)∈E , the Kirchhoff law (2.3) is a linear system of equations for the vector of pressures P = (P i ) i∈V . With the global mass conservation (2.4), the linear system (2.3) is solvable if and only if the graph with edge weights C = (C ij ) (i,j)∈E is connected [2], where only edges with positive conductivities C ij > 0 are taken into account (i.e., edges with zero conductivities are discarded). Note that the solution is unique up to an additive constant.…”

Section: Model Of Hu and Caimentioning

confidence: 99%

Auxin transport model for leaf venation

et al. 2019

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The plant hormone auxin controls many aspects of the development of plants. One striking dynamical feature is the self-organisation of leaf venation patterns which is driven by high levels of auxin within vein cells. The auxin transport is mediated by specialised membranelocalised proteins. Many venation models have been based on polarly localised efflux-mediator proteins of the PIN family. Here, we investigate a modeling framework for auxin transport with a positive feedback between auxin fluxes and transport capacities that are not necessarily polar, i.e. directional across a cell wall. Our approach is derived from a discrete graph-based model for biological transportation networks, where cells are represented by graph nodes and intercellular membranes by edges. The edges are not a-priori oriented and the direction of auxin flow is determined by its concentration gradient along the edge. We prove global existence of solutions to the model and the validity of Murray's law for its steady states. Moreover, we demonstrate with numerical simulations that the model is able connect an auxin source-sink pair with a midvein and that it can also produce branching vein patterns. A significant innovative aspect of our approach is that it allows the passage to a formal macroscopic limit which can be extended to include network growth. We perform mathematical analysis of the macroscopic formulation, showing the global existence of weak solutions for an appropriate parameter range.

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Section: Model Of Hu and Caimentioning

confidence: 99%

Auxin transport model for leaf venation

et al. 2019

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“…The corresponding fluxes, which we denote by Q ij [C], are then calculated uniquely from (7). The conductivities C ij are subject to an energy optimization and adaptation process, where, in analogy to (2), the cost functional is a sum of pumping and metabolic energies over all edges of the graph,…”

Section: Murray's Law For the Discrete Network Modelmentioning

confidence: 99%

“…Its phenomenological derivation, carried out in Ref. [2], assumes that the network domain Ω ⊂ R d is occupied by a porous medium with the permeability tensor…”

Section: Murray's Law For the Phenomenological Continuum Modelmentioning

confidence: 99%

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Murray’s law for discrete and continuum models of biological networks

Haškovec

Markowich

Pilli

2019

Math. Models Methods Appl. Sci.

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We demonstrate the validity of Murray's law, which represents a scaling relation for branch conductivities in a transportation network, for discrete and continuum models of biological networks. We first consider discrete networks with general metabolic coefficient and multiple branching nodes and derive a generalization of the classical 3/4-law. Next we prove an analogue of the discrete Murray's law for the continuum system obtained in the continuum limit of the discrete model on a rectangular mesh. Finally, we consider a continuum model derived from phenomenological considerations and show the validity of the Murray's law for its linearly stable steady states.

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“…In this paper we study the mesoscopic model briefly introduced in [2] as a bridge between the microscopic and macroscopic descriptions. The model has a formal Wasserstein-type gradient flow structure, constrained again by a Poisson equation.…”

Section: Introductionmentioning

confidence: 99%

A mesoscopic model of biological transportation networks

Burger

Haškovec

Markowich

et al. 2019

Communications in Mathematical Sciences

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We introduce a mesoscopic model for natural network formation processes, acting as a bridge between the discrete and continuous network approach proposed in [17]. The models are based on a common approach where the dynamics of the conductance network is subject to pressure force effects. We first study topological properties of the discrete model and we prove that if the metabolic energy consumption term is concave with respect to the conductivities, the optimal network structure is a tree (i.e., no loops are present). We then analyze various aspects of the mesoscopic modeling approach, in particular its relation to the discrete model and its stationary solutions, including discrete network solutions. Moreover, we present an alternative formulation of the mesoscopic model that avoids the explicit presence of the pressure in the energy functional.AMSC: 35B36; 92C42; 35K55; 49J20

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Continuum Modeling of Biological Network Formation

Cited by 26 publications

References 42 publications

Auxin transport model for leaf venation

Auxin transport model for leaf venation

Murray’s law for discrete and continuum models of biological networks

A mesoscopic model of biological transportation networks

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