SIAM Journal on Applied Mathematics, Volume 81, Issue 3, Page 741-764, January 2021.
Many foraging microorganisms rely upon cellular transport networks to deliver nutrients, fluid, and organelles between different parts of the organism. Networked organisms ranging from filamentous fungi to slime molds demonstrate a remarkable ability to mix or disperse molecules and organelles in their transport media. Here we introduce mathematical tools to analyze the structure of energy efficient transport networks that maximize mixing and sending signals originating from and arriving at each node. We define two types of entropy on flows to quantify mixing and develop numerical algorithms to optimize the combination of entropy and energy on networks, given constraints on the amount of available material. We present an in-depth exploration of optimal single source-sink networks on finite triangular grids, a fundamental setting for optimal transport networks in the plane. Using numerical simulations and rigorous proofs, we show that if the constraint on conductances is strict, the optimal networks are paths of every possible length. If the constraint is relaxed, our algorithm produces loopy networks that fan out at the source and pour back into a single path that flows to the sink. Taken together, our results expand the class of optimal transportation networks that can be compared with real biological data and highlight how real network morphologies may be shaped by trade-offs between transport efficiency and the need to mix the transported matter.

中文翻译：

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运输网络中的最佳混合：数值优化和分析

SIAM应用数学杂志，第81卷，第3期，第741-764页，2021年1月。
许多觅食微生物都依靠细胞运输网络在生物的不同部位之间输送养分，体液和细胞器。从丝状真菌到粘液霉菌的网络化生物体均具有在其运输介质中混合或分散分子和细胞器的出色能力。在这里，我们介绍数学工具来分析节能运输网络的结构，该结构可最大程度地混合和发送源自每个节点的信号并到达每个节点的信号。在给定可用材料数量的限制下，我们定义了两种类型的流动熵来量化混合，并开发了数值算法来优化网络上的熵和能量的组合。我们对有限三角形网格上的最佳单源汇网络进行了深入探索，飞机上最佳运输网络的基本设置。使用数值模拟和严格的证明，我们表明，如果对电导的约束严格，则最佳网络将是所有可能长度的路径。如果放宽了约束，我们的算法将产生循环网络，这些循环网络在源处呈扇形散开，然后倒回到流入宿的单个路径中。两者合计，我们的结果扩展了可与真实生物数据进行比较的最佳运输网络类别，并强调了运输效率与混合运输物质的需求之间的取舍可能会形成真实的网络形态。如果放宽了约束，我们的算法将产生循环网络，这些循环网络在源处呈扇形散开，然后倒回到流入宿的单个路径中。两者合计，我们的结果扩展了可与真实生物数据进行比较的最佳运输网络类别，并强调了运输效率与混合运输物质的需求之间的取舍可能会形成真实的网络形态。如果放宽了约束，我们的算法将产生循环网络，这些循环网络在源处呈扇形散开，然后倒回到流入宿的单个路径中。两者合计，我们的结果扩展了可与真实生物数据进行比较的最佳运输网络类别，并强调了运输效率与混合运输物质的需求之间的取舍可能会形成真实的网络形态。