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Trade-offs and design principles in the spatial organization of catalytic particles

Nature Physics volume 18, pages 203–211 (2022)Cite this article

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Abstract

Catalytic particles are spatially organized in a number of biological systems across different length scales, from enzyme complexes to metabolically coupled cells. Despite operating on different scales, these systems all feature localized reactions involving partially hindered diffusive transport, which is determined by the collective arrangement of the catalysts. Yet it remains largely unexplored how different arrangements affect the interplay between the reaction and transport dynamics, which ultimately determines the flux through the reaction pathway. Here we show that two fundamental trade-offs arise, the first between efficient inter-catalyst transport and the depletion of substrate, and the second between steric confinement of intermediate products and the accessibility of catalysts to substrate. We use a model reaction pathway to characterize the general design principles for the arrangement of catalysts that emerge from the interplay of these trade-offs. We find that the question of optimal catalyst arrangements generalizes the well-known Thomson problem of electrostatics.

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Fig. 1: Schematic illustration of the model.
Fig. 2: Properties of the pathway flux \({j}_{{{{\mathcal{P}}}}}\) for randomly positioned catalysts (two-dimensional system with absorbing boundary for intermediates, parameters chosen symmetrically with \({N}_{{{{\mathcal{A}}}}}={N}_{{{{\mathcal{B}}}}}=N\) and \({\alpha }_{{{{\mathcal{A}}}}}={\alpha }_{{{{\mathcal{B}}}}}=\alpha\)).
Fig. 3: Comparison of different localization strategies.
Fig. 4: Trade-off between substrate shielding and confinement of intermediates.
Fig. 5: Optimal geometries of complexes with a single \({{{\mathcal{A}}}}\) catalyst (blue) surrounded by different numbers \({N}_{{{{\mathcal{B}}}}}\) of \({{{\mathcal{B}}}}\) catalysts (orange), in 2D and 3D.
Fig. 6: Analysis of the string and sheet localization strategies studied experimentally in ref. 50.

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Data availability

The numerical data generated for this study are available from the corresponding author upon reasonable request.

Code availability

The codes used to generate the numerical data that supports the findings of this study are available on GitHub at: https://github.com/gerland-group/flux_of_cat_arrangements

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Acknowledgements

We thank E. Frey, D. Nelson and A. Walczak for useful discussions, and B. Altaner and G. Giunta for comments on the initial manuscript. This work was supported by the German Research Foundation (DFG) through SFB 1032 (project ID 201269156, to U.G.) and the Excellence Cluster ORIGINS under Germany’s Excellence Strategy (EXC-2094-390783311, to U.G.). F.H. was supported by a DFG fellowship through the Graduate School of Quantitative Biosciences Munich.

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  1. Physik-Department, Technische Universität München, Garching, Germany

    Florian Hinzpeter, Filipe Tostevin, Alexander Buchner & Ulrich Gerland

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  1. Florian Hinzpeter
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All authors designed the research. F.H. performed the research. F.H., F.T. and U.G. analysed the results and wrote the paper.

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Correspondence to Ulrich Gerland.

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Hinzpeter, F., Tostevin, F., Buchner, A. et al. Trade-offs and design principles in the spatial organization of catalytic particles. Nat. Phys. 18, 203–211 (2022). https://doi.org/10.1038/s41567-021-01444-4

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