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Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction-diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation., (© 2024. The Author(s).)

Fingerprints are complex and individually unique patterns in the skin. Established prenatally, the molecular and cellular mechanisms that guide fingerprint ridge formation and their intricate arrangements are unknown. Here we show that fingerprint ridges are epithelial structures that undergo a truncated hair follicle developmental program and fail to recruit a mesenchymal condensate. Their spatial pattern is established by a Turing reaction-diffusion system, based on signaling between EDAR, WNT, and antagonistic BMP pathways. These signals resolve epithelial growth into bands of focalized proliferation under a precociously differentiated suprabasal layer. Ridge formation occurs as a set of waves spreading from variable initiation sites defined by the local signaling environments and anatomical intricacies of the digit, with the propagation and meeting of these waves determining the type of pattern that forms. Relying on a dynamic patterning system triggered at spatially distinct sites generates the characteristic types and unending variation of human fingerprint patterns., Competing Interests: Declaration of interests The authors declare no competing interests., (Copyright © 2023 The Author(s). Published by Elsevier Inc. All rights reserved.)

We revisit the description of reaction-diffusion phenomena within nonequilibrium thermodynamics and investigate the role of a nonstandard splitting of the entropy balance into the entropy production and the divergence of entropy flux. As previously reported by Pavelka et al. (Int J Eng Sci 78:192-217, 2014), a new term is identified following from the kinetic energy of diffusion. This newly appearing term acts as a thermodynamic force driving the reaction kinetics. Using the standard constitutive relations within the linear nonequilibrium thermodynamics, the governing equations for a reaction-diffusion problem in a two-species system are derived. They turn out to be linked to Burgers' equation. It is shown that the onset of stability is not altered, but a non-periodic pattern can emerge. The latter follows from the relation of the governing equation to Burger's equation with a source term. Hence, transients formed by glued and merging parabolic profiles are expected to appear at least in certain parameter regimes. We explore the significance of this effect and observe that for a comparable magnitude of the diffusion and of the new term stemming from the kinetic energy of diffusion, the solution is expected to be linked to the saw-tooth like solution to Burger's equation rather than to the eigenmodes of the Laplacian. We conclude that the reaction-diffusion model proposed by Turing is robust to the addition of this effect of the kinetic energy of diffusion, at least when this new term is sufficiently small. As the governing equations can be rewritten into the classical reaction-diffusion problem but with reaction kinetics outside of the classical law of mass action, the analysis presented in this study suggests that a yet richer behaviour of the classical reaction-diffusion problems can be expected, if nonstandard reaction kinetics are considered., (© 2021. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of 'trivial' base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Elucidating pattern forming processes is an important problem in the physical, chemical and biological sciences. Turing's contribution, after being initially neglected, eventually catalysed a huge amount of work from mathematicians, physicists, chemists and biologists aimed towards understanding how steady spatial patterns can emerge from homogeneous chemical mixtures due to the reaction and diffusion of different chemical species. While this theory has been developed mathematically and investigated experimentally for over half a century, many questions still remain unresolved. This theme issue places Turing's theory of pattern formation in a modern context, discussing the current frontiers in foundational aspects of pattern formation in reaction-diffusion and related systems. It highlights ongoing work in chemical, synthetic and developmental settings which is helping to elucidate how important Turing's mechanism is for real morphogenesis, while highlighting gaps that remain in matching theory to reality. The theme issue also surveys a variety of recent mathematical research pushing the boundaries of Turing's original theory to more realistic and complicated settings, as well as discussing open theoretical challenges in the analysis of such models. It aims to consolidate current research frontiers and highlight some of the most promising future directions. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Realistic examples of reaction-diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of 'open' reaction-diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction-diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction-diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction-diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace-Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach.

Reaction-diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal-mesenchymal coupling in development, and symmetry-breaking in cell polarization. We develop a modeling framework for bilayer reaction-diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction-diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of pattern-forming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role of stratified media in modulating pattern-forming processes in developmental biology and beyond.

We place the Landau theory of critical phenomena into the larger context of multiscale thermodynamics. The thermodynamic potentials, with which the Landau theory begins, arise as Lyapunov like functions in the investigation of the relations among different levels of description. By seeing the renormalization-group approach to critical phenomena as inseparability of levels in the critical point, we can adopt the renormalization-group viewpoint into the Landau theory and by doing it bring its predictions closer to results of experimental observations.

Pattern formation from homogeneity is well studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is non-trivial to separate observed spatial patterning due to inherent spatial heterogeneity from emergent patterning due to nonlinear instability. We employ WKBJ asymptotics to investigate Turing instabilities for a spatially heterogeneous reaction-diffusion system, and derive conditions for instability which are local versions of the classical Turing conditions. We find that the structure of unstable modes differs substantially from the typical trigonometric functions seen in the spatially homogeneous setting. Modes of different growth rates are localized to different spatial regions. This localization helps explain common amplitude modulations observed in simulations of Turing systems in heterogeneous settings. We numerically demonstrate this theory, giving an illustrative example of the emergent instabilities and the striking complexity arising from spatially heterogeneous reaction-diffusion systems. Our results give insight both into systems driven by exogenous heterogeneity, as well as successive pattern forming processes, noting that most scenarios in biology do not involve symmetry breaking from homogeneity, but instead consist of sequential evolutions of heterogeneous states. The instability mechanism reported here precisely captures such evolution, and extends Turing's original thesis to a far wider and more realistic class of systems.

The study of pattern emergence together with exploration of the exemplar Turing model is enjoying a renaissance both from theoretical and experimental perspective. Here, we implement a stability analysis of spatially dependent reaction kinetics by exploring the effect of a jump discontinuity within piecewise constant kinetic parameters, using various methods to identify and confirm the diffusion-driven instability conditions. Essentially, the presence of stability or instability in Turing models is a local property for piecewise constant kinetic parameters and, as such, may be analyzed locally. In particular, a local assessment of whether parameters are within the Turing space provides a strong indication that for a large enough region with these parameters, an instability can be induced.

Articular cartilage is a complex, anisotropic, stratified tissue with remarkable resilience and mechanical properties. It has been subject to extensive modelling as a multiphase medium, with many recent studies examining the impact of increasing detail in the representation of this tissue's fine scale structure. However, further investigation of simple models with minimal constitutive relations can nonetheless inform our understanding at the foundations of soft tissue simulation. Here, we focus on the impact of heterogeneity with regard to the volume fractions of solid and fluid within the cartilage. Once swelling pressure due to cartilage fixed charge is also present, we demonstrate that the multiphase modelling framework is substantially more complicated, and thus investigate this complexity, especially in the simple setting of a confined compression experiment. Our findings highlight the importance of locally, and thus heterogeneously, approaching pore compaction for load bearing in cartilage models, while emphasising that such effects can be represented by simple constitutive relations. In addition, simulation predictions are observed for the sensitivity of stress and displacement in the cartilage to variations in the initial state of the cartilage and thus the details of experimental protocol, once the tissue is heterogeneous. These findings are for the simplest models given only heterogeneity in volume fractions and swelling pressure, further emphasising that the complex behaviours associated with the interaction of volume fraction heterogeneity and swelling pressure are likely to persist for simulations of cartilage representations with more fine-grained structural detail of the tissue.

Any physical system can be regarded on different levels of description varying by how detailed the description is. We propose a method called Dynamic MaxEnt (DynMaxEnt) that provides a passage from the more detailed evolution equations to equations for the less detailed state variables. The method is based on explicit recognition of the state and conjugate variables, which can relax towards the respective quasi-equilibria in different ways. Detailed state variables are reduced using the usual principle of maximum entropy (MaxEnt), whereas relaxation of conjugate variables guarantees that the reduced equations are closed. Moreover, an infinite chain of consecutive DynMaxEnt approximations can be constructed. The method is demonstrated on a particle with friction, complex fluids (equipped with conformation and Reynolds stress tensors), hyperbolic heat conduction and magnetohydrodynamics.

Phenomenological coefficients in linear nonequilibrium thermodynamics have been considered to be independent, apart from restrictions due to the Onsager-Casimir reciprocal relations and the requirement to have non-negative entropy production. Recently, it has been shown that functional constraints between these coefficients may hold, restricting their dependence on state variables, especially in the case of coupled phenomena. Here we demonstrate that such restrictions require only mild assumptions on the system of interest and are, in fact, much more constraining than previously reported. Such constraints vastly reduce the set of plausible models for constitutive relations and allow for simpler experimental determinations of dependencies in coupled systems. These results may also clarify inconsistencies in the literature regarding constitutive models used that do not obey these thermodynamic constraints.

Landau damping is the tendency of solutions to the Vlasov equation towards spatially homogeneous distribution functions. The distribution functions, however, approach the spatially homogeneous manifold only weakly, and Boltzmann entropy is not changed by the Vlasov equation. On the other hand, density and kinetic energy density, which are integrals of the distribution function, approach spatially homogeneous states strongly, which is accompanied by growth of the hydrodynamic entropy. Such a behavior can be seen when the Vlasov equation is reduced to the evolution equations for density and kinetic energy density by means of the Ehrenfest reduction.

We report on an instability arising in activator-inhibitor reaction-diffusion (RD) systems with a simple spatial heterogeneity. This instability gives rise to periodic creation, translation, and destruction of spike solutions that are commonly formed due to Turing instabilities. While this behavior is oscillatory in nature, it occurs purely within the Turing space such that no region of the domain would give rise to a Hopf bifurcation for the homogeneous equilibrium. We use the shadow limit of the Gierer-Meinhardt system to show that the speed of spike movement can be predicted from well-known asymptotic theory, but that this theory is unable to explain the emergence of these spatiotemporal oscillations. Instead, we numerically explore this system and show that the oscillatory behavior is caused by the destabilization of a steady spike pattern due to the creation of a new spike arising from endogeneous activator production. We demonstrate that on the edge of this instability, the period of the oscillations goes to infinity, although it does not fit the profile of any well-known bifurcation of a limit cycle. We show that nearby stationary states are either Turing unstable or undergo saddle-node bifurcations near the onset of the oscillatory instability, suggesting that the periodic motion does not emerge from a local equilibrium. We demonstrate the robustness of this spatiotemporal oscillation by exploring small localized heterogeneity and showing that this behavior also occurs in the Schnakenberg RD model. Our results suggest that this phenomenon is ubiquitous in spatially heterogeneous RD systems, but that current tools, such as stability of spike solutions and shadow-limit asymptotics, do not elucidate understanding. This opens several avenues for further mathematical analysis and highlights difficulties in explaining how robust patterning emerges from Turing's mechanism in the presence of even small spatial heterogeneity.

Two theories address the origin of repeating patterns, such as hair follicles, limb digits, and intestinal villi, during development. The Turing reaction-diffusion system posits that interacting diffusible signals produced by static cells first define a prepattern that then induces cell rearrangements to produce an anatomical structure. The second theory, that of mesenchymal self-organisation, proposes that mobile cells can form periodic patterns of cell aggregates directly, without reference to any prepattern. Early hair follicle development is characterised by the rapid appearance of periodic arrangements of altered gene expression in the epidermis and prominent clustering of the adjacent dermal mesenchymal cells. We assess the contributions and interplay between reaction-diffusion and mesenchymal self-organisation processes in hair follicle patterning, identifying a network of fibroblast growth factor (FGF), wingless-related integration site (WNT), and bone morphogenetic protein (BMP) signalling interactions capable of spontaneously producing a periodic pattern. Using time-lapse imaging, we find that mesenchymal cell condensation at hair follicles is locally directed by an epidermal prepattern. However, imposing this prepattern's condition of high FGF and low BMP activity across the entire skin reveals a latent dermal capacity to undergo spatially patterned self-organisation in the absence of epithelial direction. This mesenchymal self-organisation relies on restricted transforming growth factor (TGF) β signalling, which serves to drive chemotactic mesenchymal patterning when reaction-diffusion patterning is suppressed, but, in normal conditions, facilitates cell movement to locally prepatterned sources of FGF. This work illustrates a hierarchy of periodic patterning modes operating in organogenesis.

Reaction-diffusion models following the original idea of Turing are widely applied to study the propensity of a system to develop a pattern. To this end, an asymptotic analysis is typically performed via the so-called dispersion relation that relates the spectral properties of a spatial operator (diffusion) to the temporal behaviour of the whole initial-boundary value reaction-diffusion problem. Here, we amend this approach by studying the transient growth due to non-normality that can also lead to a pattern development in non-linear systems. We conclude by identification of the significance of this transient growth and by assessing the plausibility of the standard spectral approach. Particularly, the non-normality-induced patterns are possible but require fine parameter tuning.

A diffusively driven instability has been hypothesized as a mechanism to drive spatial self-organization in biological systems since the seminal work of Turing. Such systems are often considered on a growing domain, but traditional theoretical studies have only treated the domain size as a bifurcation parameter, neglecting the system non-autonomy. More recently, the conditions for a diffusively driven instability on a growing domain have been determined under stringent conditions, including slow growth, a restriction on the temporal interval over which the prospect of an instability can be considered and a neglect of the impact that time evolution has on the stability properties of the homogeneous reference state from which heterogeneity emerges. Here, we firstly relax this latter assumption and observe that the conditions for the Turing instability are much more complex and depend on the history of the system in general. We proceed to relax all the above constraints, making analytical progress by focusing on specific examples. With faster growth, instabilities can grow transiently and decay, making the prediction of a prospective Turing instability much more difficult. In addition, arbitrarily high spatial frequencies can destabilize, in which case the continuum approximation is predicted to break down.

Thermodynamic fluxes (diffusion fluxes, heat flux, etc.) are often proportional to thermodynamic forces (gradients of chemical potentials, temperature, etc.) via the matrix of phenomenological coefficients. Onsager's relations imply that the matrix is symmetric, which reduces the number of unknown coefficients is reduced. In this article we demonstrate that for a class of nonequilibrium thermodynamic models in addition to Onsager's relations the phenomenological coefficients must share the same functional dependence on the local thermodynamic state variables. Thermodynamic models and experimental data should be validated through consistency with the functional constraint. We present examples of coupled heat and mass transport (thermodiffusion) and coupled charge and mass transport (electro-osmotic drag). Additionally, these newly identified constraints further reduce the number of experiments needed to describe the phenomenological coefficient.

Reduction of a mesoscopic level to a level with fewer details is made by the time evolution during which the entropy increases. An extension of a mesoscopic level is a construction of a level with more details. In particular, we discuss extensions in which extra state variables are found in the vector fields appearing on the level that we want to extend. Reductions, extensions, and compatibility relations among them are formulated first in an abstract setting and then illustrated in specific mesoscopic theories.

Periprosthetic bone loss in response to total hip arthroplasty is a serious complication compromising patient's life quality as it may cause the premature failure of the implant. Stress shielding as a result of an uneven load sharing between the hip implant and the bone is a key factor leading to bone density decrease. A number of composite hip implants have been designed so far to improve load sharing characteristics. However, they have rarely been investigated from the bone remodeling point of view to predict a long-term response. This is the first study that employed a mechano-biochemical model, which considers the coupling effect between mechanical loading and bone biochemistry, to investigate bone remodeling after composite hip implantation. In this study, periprosthetic bone remodeling in the presence of Carbon fiber polyamide 12 (CF/PA12), CoCrMo and Ti alloy implants was predicted and compared. Our findings revealed that the most significant periprosthetic bone loss in response to metallic implants occurs in Gruen zone 7 (-43% with CoCrMo; -35% with Ti) and 6 (-40% with CoCrMo; -29% with Ti), while zone 4 has the lowest bone density decrease with all three implants (-9%). Also, the results showed that in terms of bone remodeling, the composite hip implant is more advantageous over the metallic ones as it provides a more uniform density change across the bone and induces less stress shielding which consequently results in a lower post-operative bone loss (-9% with CF/PA12 implant compared to -27% and -21% with CoCrMo and Ti alloy implants, respectively)., (Copyright © 2014 Elsevier Ltd. All rights reserved.)

Bone remodelling is a fundamental biological process that controls bone microrepair, adaptation to environmental loads and calcium regulation among other important processes. It is not surprising that bone remodelling has been subject of intensive both experimental and theoretical research. In particular, many mathematical models have been developed in the last decades focusing in particular aspects of this complicated phenomenon where mechanics, biochemistry and cell processes strongly interact. In this paper, we present a new model that combines most of these essential aspects in bone remodelling with especial focus on the effect of the mechanical environment into the biochemical control of bone adaptation mainly associated to the well known RANKL-RANK-OPG pathway. The predicted results show a good correspondence with experimental and clinical findings. For example, our results indicate that trabecular bone is more severely affected both in disuse and disease than cortical bone what has been observed in osteoporotic bones. In future, the methodology proposed would help to new therapeutic strategies following the evolution of bone tissue distribution in osteoporotic patients.

The general equation of nonequilibrium reversible-irreversible coupling (GENERIC) is studied in light of time-reversal transformation. It is shown that Onsager-Casimir reciprocal relations are implied by GENERIC in the near-equilibrium regime. A general structure which gives the reciprocal relations but which is valid also far from equilibrium is identified, and Onsager-Casimir reciprocal relations are generalized to far-from-equilibrium regime in this sense. Moreover, reversibility and irreversibility are carefully discussed and the results are illustrated in Hamiltonian dynamics, classical hydrodynamics, classical irreversible thermodynamics, the quantum master equation, and the Boltzmann equation.

In the human, placental structure is closely related to placental function and consequent pregnancy outcome. Studies have noted abnormal placental shape in small-for-gestational-age infants which extends to increased lifetime risk of cardiovascular disease. The origins and determinants of placental shape are incompletely understood and are difficult to study in vivo. In this paper, we model the early development of the human placenta, based on the hypothesis that this is driven by a chemoattractant effect emanating from proximal spiral arteries in the decidua. We derive and explore a two-dimensional stochastic model, and investigate the effects of loss of spiral arteries in regions near to the cord insertion on the shape of the placenta. This model demonstrates that disruption of spiral arteries can exert profound effects on placental shape, particularly if this is close to the cord insertion. Thus, placental shape reflects the underlying maternal vascular bed. Abnormal placental shape may reflect an abnormal uterine environment, predisposing to pregnancy complications. Through statistical analysis of model placentas, we are able to characterize the probability that a given placenta grew in a disrupted environment, and even able to distinguish between different disruptions.

Periprosthetic bone loss following total hip arthroplasty (THA) is a serious concern leading to the premature failure of prosthetic implant. Therefore, investigating bone remodeling in response to hip arthroplasty is of paramount for the purpose of designing long lasting prostheses. In this study, a thermodynamic-based theory, which considers the coupling between the mechanical loading and biochemical affinity as stimulus for bone formation and resorption, was used to simulate the femoral density change in response to THA. The results of the numerical simulations using 3D finite element analysis revealed that in Gruen zone 7, after remarkable postoperative bone loss, the bone density started recovering and got stabilized after 9% increase. The most significant periprosthetic bone loss was found in Gruen zone 7 (-17.93%) followed by zone 1 (-13.77%). Conversely, in zone 4, bone densification was observed (+4.63%). The results have also shown that the bone density loss in the posterior region of the proximal metaphysis was greater than that in the anterior side. This study provided a quantitative figure for monitoring the distribution variation of density throughout the femoral bone. The predicted bone density distribution before and after THA agree well with the bone morphology and previous results from the literature.

Mechano-chemical coupling has been recently recognised as an important effect in various systems as chemical reactivity can be controlled through an applied mechanical loading. Namely, Belousov-Zhabotinskii reactions in polymer gels exhibit self-sustained oscillations and have been identified to be reasonably controllable and definable to the extent that they can be harnessed to perform mechanical work at specific locations. In this paper, we use our theoretical work of nonlinear mechano-chemical coupling and investigate the possibility of providing an explanation of phenomena found in experimental research by means of this theory. We show that mechanotransduction occurs as a response to both static and dynamic mechanical stimulation, e.g., volume change and its rate, as observed experimentally and discuss the difference of their effects on oscillations. Plausible values of the quasi-stoichiometric parameter f of Oregonator model are estimated together with its dependence on mechanical stimulation. An increase in static loading, e.g., pressure, is predicted to have stimulatory effect whereas dynamic loading, e.g., rate of volume change, is predicted to be stimulatory only up to a certain threshold. Further, we offer a physically consistent explanation of the observed phenomena why some Belousov-Zhabotinskii gels require an additional mechanical stimulation to show emergence of oscillation or why "revival" of oscillations in Belousov-Zhabotinskii reactions is possible together with indications for further experimental setups.

Motivated by biological applications (e.g., bone tissue development and regeneration) we investigate coupling between mesoscopic mechanics and chemical kinetics. Governing equations of both dynamical systems are first written in a form expressing manifestly their compatibility with microscopic mechanics and thermodynamics. The same form is then required from governing equations of the coupled dynamics. The main result of the paper is an admissible form of the coupled dynamics.

Understanding the mechanisms governing and regulating self-organisation in the developing embryo is a key challenge that has puzzled and fascinated scientists for decades. Since its conception in 1952 the Turing model has been a paradigm for pattern formation, motivating numerous theoretical and experimental studies, though its verification at the molecular level in biological systems has remained elusive. In this work, we consider the influence of receptor-mediated dynamics within the framework of Turing models, showing how non-diffusing species impact the conditions for the emergence of self-organisation. We illustrate our results within the framework of hair follicle pre-patterning, showing how receptor interaction structures can be constrained by the requirement for patterning, without the need for detailed knowledge of the network dynamics. Finally, in the light of our results, we discuss the ability of such systems to pattern outside the classical limits of the Turing model, and the inherent dangers involved in model reduction.

The aim of this article was to present a new thermodynamic-based model for bone remodeling which is able to predict the functional adaptation of bone in response to changes in both mechanical and biochemical environments. The model was based on chemical kinetics and irreversible thermodynamic principles, in which bone is considered as a self-organizing system that exchanges matter, energy and entropy with its surroundings. The governing equations of the mathematical model have been numerically solved using Matlab software and implemented in ANSYS software using the Finite Element Method. With the aid of this model, the whole inner structure of bone was elucidated. The current model suggested that bone remodeling was a dynamic process which was driven by mechanical loading, metabolic factors and other external contributions. The model clearly indicated that in the absence of mechanical stimulus, the bone was not completely resorbed and reaches a new steady state after about 50% of bone loss. This finding agreed with previous clinical studies. Furthermore, results of virtual computations of bone density in a composite femur showed the development of a dense cortical bone around the medullary canal and a dense trabeculae bone between the femoral head and the calcar region of the medial cortex due to compressive stresses. The comparison of the predicted bone density with the structure of the proximal femur obtained from X-rays and using strain energy density gave credibility to the current model., (Copyright 2010 Wiley Periodicals, Inc. J Biomed Mater Res Part A, 2010.)

Understanding of the bone remodelling process has considerably increased during the last 20 years. Since the ability to simulate (and predict) the effects of bone remodelling offers substantial insights, several models have been proposed to describe this phenomenon. The strength of the presented model is that it includes biochemical control factors (e.g., the necessity of cell-to-cell contact, which is mediated by the RANKL-RANK-OPG chain during osteoclastogenesis) and mechanical stimulation, the governing equations are derived from interaction kinetics (e.g., mass is preserved in running reactions), and the parameters are measurable. Behaviour of the model is in accordance with experimental and clinical observations, such as the role of dynamic loading, the inhibitory effect of dynamic loading on osteoclastogenesis, the observation that polykaryon osteoclasts are activated and formed by a direct cell-to-cell contact, and the correct concentrations of osteoblasts, osteoclasts, and osteocytes. The model does not yet describe the bone remodelling process in complete detail, but the implemented simplifications describe the key features and further details of control mechanisms may be added.

The aim of this work is to address the question of what constitutes a mechanical stimulation of biochemical reactions in general and further to compare the importance of the two possible mechanical stimulations: shear rate and the rate of volume variation. Using linear nonequilibrium thermodynamics, the Curie principle (the relation for coupling phenomena) is retrieved for a phenomenological relation for a scalar flux in an isotropic system. From these phenomenological relations for the rate of chemical reaction, it is established that the only scalar quantity related to the rate of deformation tensor D that cannot be neglected is the rate of volume variation D((1)). This leads us to the conclusion that, although tissues are exposed to all variety of mechanical factors: straining, shear, pressure, and even dynamic electric fields, the volume variation rate D((1)) is the most important mechanical stimulus driving the processes in them.

This paper offers a theoretical explanation of the coupling effect phenomenon between mechanical loading and chemical reactions based on linear nonequilibrium thermodynamics and also discusses the classical method of obtaining restrictions on the phenomenological coefficients. The question whether static or dynamic loading influences biochemical processes is addressed-the necessity of dynamic loading as a stimulatory mechanism is shown. Further, the presented paper suggests that chemical and mechanical processes do not only facilitate or support one another but they may also play a triggering role for the other coupled process-some biochemical processes may need mechanical stimulation to run and vice versa as well-chemical reactions may provide energy for some mechanical processes. As an example, a detailed analysis of a model for controlled autocatalytic reproduction is presented, where the coupling effect, i.e. the influence of dynamic loading on reaction kinetics, is demonstrated.