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Matas-Gil A, Endres RG. Unraveling biochemical spatial patterns: Machine learning approaches to the inverse problem of stationary Turing patterns. iScience 2024; 27:109822. [PMID: 38827409 PMCID: PMC11140185 DOI: 10.1016/j.isci.2024.109822] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2023] [Revised: 03/14/2024] [Accepted: 04/24/2024] [Indexed: 06/04/2024] Open
Abstract
The diffusion-driven Turing instability is a potential mechanism for spatial pattern formation in numerous biological and chemical systems. However, engineering these patterns and demonstrating that they are produced by this mechanism is challenging. To address this, we aim to solve the inverse problem in artificial and experimental Turing patterns. This task is challenging since patterns are often corrupted by noise and slight changes in initial conditions can lead to different patterns. We used both least squares to explore the problem and physics-informed neural networks to build a noise-robust method. We elucidate the functionality of our network in scenarios mimicking biological noise levels and showcase its application using an experimentally obtained chemical pattern. The findings reveal the significant promise of machine learning in steering the creation of synthetic patterns in bioengineering, thereby advancing our grasp of morphological intricacies within biological systems while acknowledging existing limitations.
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Affiliation(s)
- Antonio Matas-Gil
- Department of Life Sciences & Centre for Integrative Systems Biology and Bioinformatics, Imperial College London, London SW7 2BU, UK
| | - Robert G. Endres
- Department of Life Sciences & Centre for Integrative Systems Biology and Bioinformatics, Imperial College London, London SW7 2BU, UK
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Li B, Zhu L. Turing instability analysis and parameter identification based on optimal control and statistics method for a rumor propagation system. CHAOS (WOODBURY, N.Y.) 2024; 34:053143. [PMID: 38814676 DOI: 10.1063/5.0207411] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2024] [Accepted: 05/06/2024] [Indexed: 05/31/2024]
Abstract
This study establishes a reaction-diffusion system to capture the dynamics of rumor propagation, considering two possibilities of contact transmission. The sufficient and necessary conditions for a positive equilibrium point are provided, and the Turing instability conditions for this equilibrium point are derived. Furthermore, utilizing variational inequalities, a first-order necessary condition for parameter identification based on optimal control is established. During the numerical simulation process, the correctness of the Turing instability conditions is verified, and optimal control-based parameter identification is applied to the target pattern. Additionally, statistical methods are employed for pattern parameter identification. The identification results demonstrate that optimal control-based parameter identification exhibits higher efficiency and accuracy. Finally, both theories' parameter identification principles are extended to a small-world network, yielding consistent conclusions with continuous space.
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Affiliation(s)
- Bingxin Li
- School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China
| | - Linhe Zhu
- School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China
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Chang L, Wang X, Sun G, Wang Z, Jin Z. A time independent least squares algorithm for parameter identification of Turing patterns in reaction-diffusion systems. J Math Biol 2023; 88:5. [PMID: 38017080 DOI: 10.1007/s00285-023-02026-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2023] [Revised: 09/28/2023] [Accepted: 10/29/2023] [Indexed: 11/30/2023]
Abstract
Turing patterns arising from reaction-diffusion systems such as epidemic, ecology or chemical reaction models are an important dynamic property. Parameter identification of Turing patterns in spatial continuous and networked reaction-diffusion systems is an interesting and challenging inverse problem. The existing algorithms require huge account operations and resources. These drawbacks are amplified when apply them to reaction-diffusion systems on large-scale complex networks. To overcome these shortcomings, we present a new least squares algorithm which is rooted in the fact that Turing patterns are the stationary solutions of reaction-diffusion systems. The new algorithm is time independent, it translates the parameter identification problem into a low dimensional optimization problem even a low order linear algebra equations. The numerical simulations demonstrate that our algorithm has good effectiveness, robustness as well as performance.
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Affiliation(s)
- Lili Chang
- Complex Systems Research Center, Shanxi University, Taiyuan, 030006, China.
- Key Laboratory of Complex Systems and Data Science of Ministry of Education, Taiyuan, 030006, China.
| | - Xinyu Wang
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an, 710072, China
- School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an, 710072, China
| | - Guiquan Sun
- Key Laboratory of Complex Systems and Data Science of Ministry of Education, Taiyuan, 030006, China.
- Department of Mathematics, North University of China, Taiyuan, 030051, China.
| | - Zhen Wang
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an, 710072, China
- School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an, 710072, China
| | - Zhen Jin
- Complex Systems Research Center, Shanxi University, Taiyuan, 030006, China
- Key Laboratory of Complex Systems and Data Science of Ministry of Education, Taiyuan, 030006, China
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Schnörr D, Schnörr C. Learning system parameters from turing patterns. Mach Learn 2023; 112:3151-3190. [PMID: 37575882 PMCID: PMC10415500 DOI: 10.1007/s10994-023-06334-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2021] [Revised: 03/14/2023] [Accepted: 03/30/2023] [Indexed: 08/15/2023]
Abstract
The Turing mechanism describes the emergence of spatial patterns due to spontaneous symmetry breaking in reaction-diffusion processes and underlies many developmental processes. Identifying Turing mechanisms in biological systems defines a challenging problem. This paper introduces an approach to the prediction of Turing parameter values from observed Turing patterns. The parameter values correspond to a parametrized system of reaction-diffusion equations that generate Turing patterns as steady state. The Gierer-Meinhardt model with four parameters is chosen as a case study. A novel invariant pattern representation based on resistance distance histograms is employed, along with Wasserstein kernels, in order to cope with the highly variable arrangement of local pattern structure that depends on the initial conditions which are assumed to be unknown. This enables us to compute physically plausible distances between patterns, to compute clusters of patterns and, above all, model parameter prediction based on training data that can be generated by numerical model evaluation with random initial data: for small training sets, classical state-of-the-art methods including operator-valued kernels outperform neural networks that are applied to raw pattern data, whereas for large training sets the latter are more accurate. A prominent property of our approach is that only a single pattern is required as input data for model parameter predicion. Excellent predictions are obtained for single parameter values and reasonably accurate results for jointly predicting all four parameter values.
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Affiliation(s)
- David Schnörr
- School of Life Sciences, Imperial College, London, UK
| | - Christoph Schnörr
- Institute for Applied Mathematics, Heidelberg University, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
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Yoshinaga N, Tokuda S. Bayesian modeling of pattern formation from one snapshot of pattern. Phys Rev E 2022; 106:065301. [PMID: 36671103 DOI: 10.1103/physreve.106.065301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/12/2022] [Accepted: 11/07/2022] [Indexed: 06/17/2023]
Abstract
Partial differential equations (PDEs) have been widely used to reproduce patterns in nature and to give insight into the mechanism underlying pattern formation. Although many PDE models have been proposed, they rely on the pre-request knowledge of physical laws and symmetries, and developing a model to reproduce a given desired pattern remains difficult. We propose a method, referred to as Bayesian modeling of PDEs (BM-PDEs), to estimate the best dynamical PDE for one snapshot of a objective pattern under the stationary state without ground truth. We apply BM-PDEs to nontrivial patterns, such as quasicrystals (QCs), a double gyroid, and Frank-Kasper structures. We also generate three-dimensional dodecagonal QCs from a PDE model. This is done by using the estimated parameters for the Frank-Kasper A15 structure, which closely approximates the local structures of QCs. Our method works for noisy patterns and the pattern synthesized without the ground-truth parameters, which are required for the application toward experimental data.
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Affiliation(s)
- Natsuhiko Yoshinaga
- WPI-Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
- MathAM-OIL, AIST, Sendai 980-8577, Japan
| | - Satoru Tokuda
- MathAM-OIL, AIST, Sendai 980-8577, Japan
- Research Institute for Information Technology, Kyushu University, Kasuga 816-8580, Japan
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Veerman F, Mercker M, Marciniak-Czochra A. Beyond Turing: far-from-equilibrium patterns and mechano-chemical feedback. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2021; 379:20200278. [PMID: 34743599 DOI: 10.1098/rsta.2020.0278] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
Turing patterns are commonly understood as specific instabilities of a spatially homogeneous steady state, resulting from activator-inhibitor interaction destabilized by diffusion. We argue that this view is restrictive and its agreement with biological observations is problematic. We present two alternatives to the classical Turing analysis of patterns. First, we employ the abstract framework of evolution equations to enable the study of far-from-equilibrium patterns. Second, we introduce a mechano-chemical model, with the surface on which the pattern forms being dynamic and playing an active role in the pattern formation, effectively replacing the inhibitor. We highlight the advantages of these two alternatives vis-à-vis the classical Turing analysis, and give an overview of recent results and future challenges for both approaches. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.
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Affiliation(s)
- Frits Veerman
- University of Leiden, Mathematical Institute, Niels Bohrweg 1, Leiden 2333 CA, The Netherlands
| | - Moritz Mercker
- Institute for Applied Mathematics and Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Im Neuenheimer Feld 205, Heidelberg 69120, Germany
| | - Anna Marciniak-Czochra
- Institute for Applied Mathematics and Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Im Neuenheimer Feld 205, Heidelberg 69120, Germany
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Abstract
Reaction-diffusion systems are an intensively studied form of partial differential equation, frequently used to produce spatially heterogeneous patterned states from homogeneous symmetry breaking via the Turing instability. Although there are many prototypical "Turing systems" available, determining their parameters, functional forms, and general appropriateness for a given application is often difficult. Here, we consider the reverse problem. Namely, suppose we know the parameter region associated with the reaction kinetics in which patterning is required-we present a constructive framework for identifying systems that will exhibit the Turing instability within this region, whilst in addition often allowing selection of desired patterning features, such as spots, or stripes. In particular, we show how to build a system of two populations governed by polynomial morphogen kinetics such that the: patterning parameter domain (in any spatial dimension), morphogen phases (in any spatial dimension), and even type of resulting pattern (in up to two spatial dimensions) can all be determined. Finally, by employing spatial and temporal heterogeneity, we demonstrate that mixed mode patterns (spots, stripes, and complex prepatterns) are also possible, allowing one to build arbitrarily complicated patterning landscapes. Such a framework can be employed pedagogically, or in a variety of contemporary applications in designing synthetic chemical and biological patterning systems. We also discuss the implications that this freedom of design has on using reaction-diffusion systems in biological modelling and suggest that stronger constraints are needed when linking theory and experiment, as many simple patterns can be easily generated given freedom to choose reaction kinetics.
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Affiliation(s)
- Thomas E Woolley
- Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK.
| | - Andrew L Krause
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Eamonn A Gaffney
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
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