Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

On the theory of cell migration: durotaxis and chemotaxis

Cell migration is a fundamental element in a variety of physiological and pathological processes. Alteration of its regulatory mechanisms leads to loss of cellular adhesion and increased motility, which are critical steps in the initial stages of metastasis, before a malignant cell colonizes a distant tissue or organ. Consequently, cell migration has become the focus of intensive experimental and theoretical studies; however the understanding of many of its mechanism remains elusive. Cell migration is the result of a periodic sequence of protrusion, adhesion remodeling and contraction stages that leads to directed movement of cells towards external stimuli. The spatio-temporal coordination of these processes depends on the di erential activation of the signaling networks that regulate them at specific subcellular locations. Particularly, proteins from the family of small RhoGTPases play a central role in establishing cell polarization, setting the direction of migration, regulating the formation of adhesion sites and the generation of the forces that drive motion. Theoretical models based on an independent description of these processes have a limited capacity to predict cellular behavior observed in vitro, since their functionality depends intrinsically on the cross-regulation between their signaling pathways. This thesis presents a model of cell migration that integrates a description of force generation and cell deformation, adhesion site dynamics and RhoGTPases activation. The cell is modeled as a viscoelastic body capable of developing active traction and protrusion forces. The magnitude of stresses is determined by the activation level of the RhoGTPases, whose distribution in the cell body is described by a set of reaction-di usion equations. Adhesion sites are modeled as punctual clusters of transmembrane receptors that dynamically bind and unbind the extracellular matrix depending on the force transmitted to them and the distance with ligands on the substrate. Onthe theoretical level, the major findings concern the relationship between the topology of a crosstalk scheme and the properties, as defined in [1], inherited by the associated reaction network as a gradient sensing and regulatory system: persistent and transient polarization triggered by external gradients, adaptation to uniform stimulus, reversible polarization, multi-stimuli response and amplification. This leads to models that remain functional against the biological diversity associated to di erent cell types and matches the observed cell behaviour in Chemotaxis essays [2, 3, 4, 5]: the capacity of cells to amplify gradients, polarize without featuring Turing patterns of activation, and switch the polarization axis and the direction of migration after the source of the external stimulus is changed. The RhoGTPase model, derived on theoretical premises, challenges a long held view on the mechanisms of RhoGTPase crosstalk and suggests that the role of GDIs, GEFs and GAPs has to be revised. Recent experimental evidence supports this idea[6]. In addition, the model allows to recapitulate a continuous transition between the tear-like shape adopted by neutrophiles and the fan-like shape of keratocytes during migration [7] by varying the relative magnitudes of protrusion and contraction forces or, alternatively, the strength of RhoGTPase Crosstalk. The second mechanism represents a novel explanation of the di erent morphologies observed in migrating cells. Di erences in RhoGTPase crosstalk strength could be mediated by di erences between the activity or concentration of GEFs, GAPs and GDIs in di erent cell types; an idea that can be explored experimentally. On cell mechanosensing, a new hypothesis based on a simple physical principle is proposed as the mechanism that might explain the universal preference of cells (bar neurons) to migrate along sti ness gradients. The theory provides a simple unifying explanation to a number of recent observations on force development and growth in real time at cell Focal adhesions [8, 9, 10, 11]. The apparently conflicting results have been attributed to the di erences in experimental set-ups and cell types used, and have fueled a longstanding controversy on how cells prove the mechanical properties of the extra-cellular matrix. The predictions of the theory recapitulate these experimental observations, and its founding hypothesis can be tested experimentally. This hypothesis directly suggests the mechanism that could explain the preference of cells to migrate along sti ness gradients, and for the first time, a plausible biological function for its existence. This phenomenon is known as Durotaxis, and its abnormal regulation has been associated to the malignant behaviour of cancer cells. &nbsp

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

On the theory of cell migration: durotaxis and chemotaxis

Cell migration is a fundamental element in a variety of physiological and pathological processes. Alteration of its regulatory mechanisms leads to loss of adhesion and increased motility, critical steps in the initial stages of metastasis. Consequently, cell migration has become the focus of intensive experimental and theoretical studies; however the understanding many of its mechanisms remains elusive. Cell migration is the result of a periodic sequence of protrusion, adhesion remodeling and contraction stages that leads to directed movement towards external stimuli. The spatio-temporal coordination of these processes depends on the activation of the signaling networks that regulate them at specific subcellular locations. Particularly, the family of small RhoGTPases plays a central role in regulating cell polarization, the formation of adhesion sites and the generation of the forces that drive motion. Theoretical models based on an independent description of these processes have a limited capacity to predict cellular behavior observed in vitro, since their functionality depends on the cross-regulation between their signaling pathways. This thesis presents a model of cell migration that integrates a description of force generation and cell deformation, adhesion site dynamics and RhoGTPases activation. The cell is modeled as a viscoelastic body capable of developing traction and protrusion forces. The forces are determined by the activation level of the RhoGTPases, whose distribution in the cell is described by a set of reaction-diffusion equations. Adhesion sites are modeled as punctual clusters of transmembrane receptors that dynamically bind and unbind the extracellular matrix depending on the force transimtted to them and the distance with ligands coating the substrate. On the theoretical level, the major findings relate the topology of a Crosstalk Scheme and the properties inherited by the associated reaction network as a gradient sensing and regulatory system: reversible polarization, adaptation to uniform stimulus, multi-stimuli response and amplification. Models formulated according to these principles remain functional against the biological diversity associated to different cell types and match the observed behavior in Chemotaxis essays: the capacity of cells to detect shallow gradients, polarize without featuring Turing patterns of activation, and switch the direction of migration after the stimulus source is changed. The biological implications challenge a long held view on the mechanisms of RhoGTPase crosstalk and suggests that the role of GDIs, GEFs and GAPs has to be revised, as supported by recent experimental evidence. In addition, the model recapitulates a continuous transition from the tear-like shape adopted by neutrophiles to the fan-like shape of keratocytes during migration by varying the magnitudes of protrusion and contraction forces or, alternatively, the strength of RhoGTPase Crosstalk. The second mechanism represents a novel explanation of the different morphologies observed in migrating cells. On cell mechanosensing, a new hypothesis is proposed to explain how cells sense the mechanical properties of the ECM. The hypothesis provides a unifying explanation to apparently conflicting observations on force development and growth in real time at cell Focal adhesions, previously attributed to differences in experimental set-ups or cell types studied. An interpretation for the observed relationships between polarization time, migration speed, mechano-sensing limits and substrate rigidity follows from this hypothesis. Further, the theory directly suggests the currently unknown mechanisms that could explain the universal preference of cells (bar neurons) to migrate along stiffness gradients, and for the first time, a plausible biological function for the existence of this phenomenon. It is known as Durotaxis, and its abnormal regulation has been associated to the malignant behaviour of cancer cells