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Lacalli TC. Patterning, From Conifers to Consciousness: Turing's Theory and Order From Fluctuations. Front Cell Dev Biol 2022; 10:871950. [PMID: 35592249 PMCID: PMC9111979 DOI: 10.3389/fcell.2022.871950] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2022] [Accepted: 03/11/2022] [Indexed: 11/19/2022] Open
Abstract
This is a brief account of Turing's ideas on biological pattern and the events that led to their wider acceptance by biologists as a valid way to investigate developmental pattern, and of the value of theory more generally in biology. Periodic patterns have played a key role in this process, especially 2D arrays of oriented stripes, which proved a disappointment in theoretical terms in the case of Drosophila segmentation, but a boost to theory as applied to skin patterns in fish and model chemical reactions. The concept of "order from fluctuations" is a key component of Turing's theory, wherein pattern arises by selective amplification of spatial components concealed in the random disorder of molecular and/or cellular processes. For biological examples, a crucial point from an analytical standpoint is knowing the nature of the fluctuations, where the amplifier resides, and the timescale over which selective amplification occurs. The answer clarifies the difference between "inelegant" examples such as Drosophila segmentation, which is perhaps better understood as a programmatic assembly process, and "elegant" ones expressible in equations like Turing's: that the fluctuations and selection process occur predominantly in evolutionary time for the former, but in real time for the latter, and likewise for error suppression, which for Drosophila is historical, in being lodged firmly in past evolutionary events. The prospects for a further extension of Turing's ideas to the complexities of brain development and consciousness is discussed, where a case can be made that it could well be in neuroscience that his ideas find their most important application.
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Ceccarelli AS, Borges A, Chara O. Size matters: tissue size as a marker for a transition between reaction-diffusion regimes in spatio-temporal distribution of morphogens. ROYAL SOCIETY OPEN SCIENCE 2022; 9:211112. [PMID: 35116146 PMCID: PMC8790355 DOI: 10.1098/rsos.211112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 06/30/2021] [Accepted: 12/22/2021] [Indexed: 06/14/2023]
Abstract
The reaction-diffusion model constitutes one of the most influential mathematical models to study distribution of morphogens in tissues. Despite its widespread use, the effect of finite tissue size on model-predicted spatio-temporal morphogen distributions has not been completely elucidated. In this study, we analytically investigated the spatio-temporal distributions of morphogens predicted by a reaction-diffusion model in a finite one-dimensional domain, as a proxy for a biological tissue, and compared it with the solution of the infinite-domain model. We explored the reduced parameter, the tissue length in units of a characteristic reaction-diffusion length, and identified two reaction-diffusion regimes separated by a crossover tissue size estimated in approximately three characteristic reaction-diffusion lengths. While above this crossover the infinite-domain model constitutes a good approximation, it breaks below this crossover, whereas the finite-domain model faithfully describes the entire parameter space. We evaluated whether the infinite-domain model renders accurate estimations of diffusion coefficients when fitted to finite spatial profiles, a procedure typically followed in fluorescence recovery after photobleaching (FRAP) experiments. We found that the infinite-domain model overestimates diffusion coefficients when the domain is smaller than the crossover tissue size. Thus, the crossover tissue size may be instrumental in selecting the suitable reaction-diffusion model to study tissue morphogenesis.
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Affiliation(s)
- Alberto S. Ceccarelli
- Systems Biology Group (SysBio), Institute of Physics of Liquids and Biological Systems (IFLySIB), National Scientific and Technical Research Council (CONICET), University of La Plata, La Plata, Argentina
| | - Augusto Borges
- Systems Biology Group (SysBio), Institute of Physics of Liquids and Biological Systems (IFLySIB), National Scientific and Technical Research Council (CONICET), University of La Plata, La Plata, Argentina
- Research Unit of Sensory Biology & Organogenesis, Helmholtz Zentrum München, Munich, Germany
- Graduate School of Quantitative Biosciences (QBM), Munich, Germany
| | - Osvaldo Chara
- Systems Biology Group (SysBio), Institute of Physics of Liquids and Biological Systems (IFLySIB), National Scientific and Technical Research Council (CONICET), University of La Plata, La Plata, Argentina
- Center for Information Services and High Performance Computing (ZIH), Technische Universität Dresden, Dresden, Germany
- Instituto de Tecnología, Universidad Argentina de la Empresa (UADE), Buenos Aires, Argentina
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Aguirre-Tamaral A, Guerrero I. Improving the understanding of cytoneme-mediated morphogen gradients by in silico modeling. PLoS Comput Biol 2021; 17:e1009245. [PMID: 34343167 PMCID: PMC8362982 DOI: 10.1371/journal.pcbi.1009245] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2021] [Revised: 08/13/2021] [Accepted: 07/03/2021] [Indexed: 01/23/2023] Open
Abstract
Morphogen gradients are crucial for the development of organisms. The biochemical properties of many morphogens prevent their extracellular free diffusion, indicating the need of an active mechanism for transport. The involvement of filopodial structures (cytonemes) has been proposed for morphogen signaling. Here, we describe an in silico model based on the main general features of cytoneme-meditated gradient formation and its implementation into Cytomorph, an open software tool. We have tested the spatial and temporal adaptability of our model quantifying Hedgehog (Hh) gradient formation in two Drosophila tissues. Cytomorph is able to reproduce the gradient and explain the different scaling between the two epithelia. After experimental validation, we studied the predicted impact of a range of features such as length, size, density, dynamics and contact behavior of cytonemes on Hh morphogen distribution. Our results illustrate Cytomorph as an adaptive tool to test different morphogen gradients and to generate hypotheses that are difficult to study experimentally. Graded distribution of signaling molecules (morphogens) is crucial for the development of organisms. Signaling membrane protrusions, called Cytonemes, have been experimentally demonstrated to be involved in morphogen transport and reception. Here, we have developed an in silico model for gradient formation based on key features of cytoneme mediated signaling. We have also implemented the model into an open software tool we named Cytomorph, and validated it by comparing its simulations with experimental data obtained from Hedgehog morphogen distribution. Finally, we have generated in silico predictions for the impact of different cytoneme features such as length, size, density, dynamics and contact behavior. Our results show that Cytomorph is an adaptive tool that can facilitate the study of other cytoneme-dependent morphogen gradients, besides being able to generate hypotheses about aspects that remain elusive to experimental approaches.
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Affiliation(s)
- Adrián Aguirre-Tamaral
- Tissue and Organ Homeostasis, Centro de Biología Molecular "Severo Ochoa" (CSIC-UAM), Universidad Autónoma de Madrid, Cantoblanco, Madrid, Spain
- * E-mail: (AA-T); (IG)
| | - Isabel Guerrero
- Tissue and Organ Homeostasis, Centro de Biología Molecular "Severo Ochoa" (CSIC-UAM), Universidad Autónoma de Madrid, Cantoblanco, Madrid, Spain
- * E-mail: (AA-T); (IG)
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Kang HW, Erban R. Multiscale Stochastic Reaction-Diffusion Algorithms Combining Markov Chain Models with Stochastic Partial Differential Equations. Bull Math Biol 2019; 81:3185-3213. [PMID: 31165406 PMCID: PMC6677718 DOI: 10.1007/s11538-019-00613-0] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2018] [Accepted: 05/09/2019] [Indexed: 12/21/2022]
Abstract
Two multiscale algorithms for stochastic simulations of reaction-diffusion processes are analysed. They are applicable to systems which include regions with significantly different concentrations of molecules. In both methods, a domain of interest is divided into two subsets where continuous-time Markov chain models and stochastic partial differential equations (SPDEs) are used, respectively. In the first algorithm, Markov chain (compartment-based) models are coupled with reaction-diffusion SPDEs by considering a pseudo-compartment (also called an overlap or handshaking region) in the SPDE part of the computational domain right next to the interface. In the second algorithm, no overlap region is used. Further extensions of both schemes are presented, including the case of an adaptively chosen boundary between different modelling approaches.
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Affiliation(s)
- Hye-Won Kang
- Department of Mathematics and Statistics, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250 USA
| | - Radek Erban
- Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG UK
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Kniss-James AS, Rivet CA, Chingozha L, Lu H, Kemp ML. Single-cell resolution of intracellular T cell Ca 2+ dynamics in response to frequency-based H 2O 2 stimulation. Integr Biol (Camb) 2017; 9:238-247. [PMID: 28164205 PMCID: PMC5360518 DOI: 10.1039/c6ib00186f] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022]
Abstract
Adaptive immune cells, such as T cells, integrate information from their extracellular environment through complex signaling networks with exquisite sensitivity in order to direct decisions on proliferation, apoptosis, and cytokine production. These signaling networks are reliant on the interplay between finely tuned secondary messengers, such as Ca2+ and H2O2. Frequency response analysis, originally developed in control engineering, is a tool used for discerning complex networks. This analytical technique has been shown to be useful for understanding biological systems and facilitates identification of the dominant behaviour of the system. We probed intracellular Ca2+ dynamics in the frequency domain to investigate the complex relationship between two second messenger signaling molecules, H2O2 and Ca2+, during T cell activation with single cell resolution. Single-cell analysis provides a unique platform for interrogating and monitoring cellular processes of interest. We utilized a previously developed microfluidic device to monitor individual T cells through time while applying a dynamic input to reveal a natural frequency of the system at approximately 2.78 mHz stimulation. Although our network was much larger with more unknown connections than previous applications, we are able to derive features from our data, observe forced oscillations associated with specific amplitudes and frequencies of stimuli, and arrive at conclusions about potential transfer function fits as well as the underlying population dynamics.
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Affiliation(s)
- Ariel S Kniss-James
- The Parker H. Petit Institute of Bioengineering and Bioscience, Georgia Institute of Technology, 315 Ferst Dr NW, Atlanta, GA 30332-0363, USA
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Quiñinao C, Prochiantz A, Touboul J. Local homeoprotein diffusion can stabilize boundaries generated by graded positional cues. Development 2015; 142:1860-8. [PMID: 25968317 PMCID: PMC5207310 DOI: 10.1242/dev.113688] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
Abstract
Boundary formation in the developing neuroepithelium decides on the position and size of compartments in the adult nervous system. In this study, we start from the French Flag model proposed by Lewis Wolpert, in which boundaries are formed through the combination of morphogen diffusion and of thresholds in cell responses. In contemporary terms, a response is characterized by the expression of cell-autonomous transcription factors, very often of the homeoprotein family. Theoretical studies suggest that this sole mechanism results in the formation of boundaries of imprecise shapes and positions. Alan Turing, on the other hand, proposed a model whereby two morphogens that exhibit self-activation and reciprocal inhibition, and are uniformly distributed and diffuse at different rates lead to the formation of territories of unpredictable shapes and positions but with sharp boundaries (the 'leopard spots'). Here, we have combined the two models and compared the stability of boundaries when the hypothesis of local homeoprotein intercellular diffusion is, or is not, introduced in the equations. We find that the addition of homeoprotein local diffusion leads to a dramatic stabilization of the positioning of the boundary, even when other parameters are significantly modified. This novel Turing/Wolpert combined model has thus important theoretical consequences for our understanding of the role of the intercellular diffusion of homeoproteins in the developmental robustness of and the changes that take place in the course of evolution.
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Affiliation(s)
- Cristóbal Quiñinao
- Collège de France, Centre for Interdisciplinary Research in Biology, UMR CNRS 7241/INSERM 1050, Labex MemoLife, 11 place Marcelin Berthelot, Paris 75231, France Laboratoire Jacques-Louis Lions, CNRS UMR 7598, Université Pierre et Marie Curie (UPMC) - Paris VI, 4 place Jussieu, Paris 75005, France
| | - Alain Prochiantz
- Collège de France, Centre for Interdisciplinary Research in Biology, UMR CNRS 7241/INSERM 1050, Labex MemoLife, 11 place Marcelin Berthelot, Paris 75231, France
| | - Jonathan Touboul
- Collège de France, Centre for Interdisciplinary Research in Biology, UMR CNRS 7241/INSERM 1050, Labex MemoLife, 11 place Marcelin Berthelot, Paris 75231, France INRIA Paris Rocquencourt, MYCENAE Team, Domaine de Voluceau, Le Chesnay 78153, France
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Umulis DM, Othmer HG. The role of mathematical models in understanding pattern formation in developmental biology. Bull Math Biol 2015; 77:817-45. [PMID: 25280665 PMCID: PMC4819020 DOI: 10.1007/s11538-014-0019-7] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2014] [Accepted: 09/02/2014] [Indexed: 12/11/2022]
Abstract
In a Wall Street Journal article published on April 5, 2013, E. O. Wilson attempted to make the case that biologists do not really need to learn any mathematics-whenever they run into difficulty with numerical issues, they can find a technician (aka mathematician) to help them out of their difficulty. He formalizes this in Wilsons Principle No. 1: "It is far easier for scientists to acquire needed collaboration from mathematicians and statisticians than it is for mathematicians and statisticians to find scientists able to make use of their equations." This reflects a complete misunderstanding of the role of mathematics in all sciences throughout history. To Wilson, mathematics is mere number crunching, but as Galileo said long ago, "The laws of Nature are written in the language of mathematics[Formula: see text] the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word." Mathematics has moved beyond the geometry-based model of Galileo's time, and in a rebuttal to Wilson, E. Frenkel has pointed out the role of mathematics in synthesizing the general principles in science (Both point and counter-point are available in Wilson and Frenkel in Notices Am Math Soc 60(7):837-838, 2013). We will take this a step further and show how mathematics has been used to make new and experimentally verified discoveries in developmental biology and how mathematics is essential for understanding a problem that has puzzled experimentalists for decades-that of how organisms can scale in size. Mathematical analysis alone cannot "solve" these problems since the validation lies at the molecular level, but conversely, a growing number of questions in biology cannot be solved without mathematical analysis and modeling. Herein, we discuss a few examples of the productive intercourse between mathematics and biology.
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Affiliation(s)
- David M. Umulis
- Agricultural and Biological Engineering, Weldon School of Biomedical Engineering, Purdue University, West Lafayette, IN 47907, USA
| | - Hans G. Othmer
- School of Mathematics and Digital Technology Center, University of Minnesota, Minneapolis, MN 55455, USA
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Hu J, Kang HW, Othmer HG. Stochastic analysis of reaction-diffusion processes. Bull Math Biol 2014; 76:854-94. [PMID: 23719732 PMCID: PMC3825834 DOI: 10.1007/s11538-013-9849-y] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2012] [Accepted: 04/25/2013] [Indexed: 11/25/2022]
Abstract
Reaction and diffusion processes are used to model chemical and biological processes over a wide range of spatial and temporal scales. Several routes to the diffusion process at various levels of description in time and space are discussed and the master equation for spatially discretized systems involving reaction and diffusion is developed. We discuss an estimator for the appropriate compartment size for simulating reaction-diffusion systems and introduce a measure of fluctuations in a discretized system. We then describe a new computational algorithm for implementing a modified Gillespie method for compartmental systems in which reactions are aggregated into equivalence classes and computational cells are searched via an optimized tree structure. Finally, we discuss several examples that illustrate the issues that have to be addressed in general systems.
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Affiliation(s)
- Jifeng Hu
- School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
| | - Hye-Won Kang
- Mathematical Biosciences Institute, Ohio State University, Columbus, OH, USA
| | - Hans G. Othmer
- School of Mathematics and Digital Technology Center, University of Minnesota, Minneapolis, MN 55455, USA
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Abstract
Many organisms and their constituent tissues and organs vary substantially in size but differ little in morphology; they appear to be scaled versions of a common template or pattern. Such scaling involves adjusting the intrinsic scale of spatial patterns of gene expression that are set up during development to the size of the system. Identifying the mechanisms that regulate scaling of patterns at the tissue, organ and organism level during development is a longstanding challenge in biology, but recent molecular-level data and mathematical modeling have shed light on scaling mechanisms in several systems, including Drosophila and Xenopus. Here, we investigate the underlying principles needed for understanding the mechanisms that can produce scale invariance in spatial pattern formation and discuss examples of systems that scale during development.
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Affiliation(s)
- David M Umulis
- Agricultural and Biological Engineering, Weldon School of Biomedical Engineering, Purdue University, West Lafayette, IN 47907, USA
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Zheng L, Chen M, Nie Q. External noise control in inherently stochastic biological systems. JOURNAL OF MATHEMATICAL PHYSICS 2012; 53:115616. [PMID: 23213267 PMCID: PMC3505197 DOI: 10.1063/1.4762825] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/16/2012] [Accepted: 10/03/2012] [Indexed: 05/24/2023]
Abstract
Biological systems are often subject to external noise from signal stimuli and environmental perturbations, as well as noises in the intracellular signal transduction pathway. Can different stochastic fluctuations interact to give rise to new emerging behaviors? How can a system reduce noise effects while still being capable of detecting changes in the input signal? Here, we study analytically and computationally the role of nonlinear feedback systems in controlling external noise with the presence of large internal noise. In addition to noise attenuation, we analyze derivatives of Fano factor to study systems' capability of differentiating signal inputs. We find effects of internal noise and external noise may be separated in one slow positive feedback loop system; in particular, the slow loop can decrease external noise and increase robustness of signaling with respect to fluctuations in rate constants, while maintaining the signal output specific to the input. For two feedback loops, we demonstrate that the influence of external noise mainly depends on how the fast loop responds to fluctuations in the input and the slow loop plays a limited role in determining the signal precision. Furthermore, in a dual loop system of one positive feedback and one negative feedback, a slower positive feedback always leads to better noise attenuation; in contrast, a slower negative feedback may not be more beneficial. Our results reveal interesting stochastic effects for systems containing both extrinsic and intrinsic noises, suggesting novel noise filtering strategies in inherently stochastic systems.
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Affiliation(s)
- Likun Zheng
- Department of Mathematics, University of California, Irvine, California 92697, USA ; Center for Complex Biological Systems, University of California, Irvine, California 92697, USA ; Center for Mathematical and Computational Biology, University of California, Irvine, California 92697, USA
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Affiliation(s)
- S. Barry Cooper
- School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
| | - Philip K. Maini
- School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
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