Institute of Fundamental Technological Research

Viral infection in cell culture and tissue is modeled with delay reaction-diffusion
equations. It is shown that progression of viral infection can be characterized by
the viral replication number, time-dependent viral load, and the speed of infec-
tion spreading. These three characteristics are determined through the original
model parameters including the rates of cell infection and of virus production in
the infected cells. The clinical manifestations of viral infection, depending on tis-
sue damage, correlate with the speed of infection spreading, while the infectivity
of a respiratory infection depends on the viral load in the upper respiratory tract.
Parameter determination from the experiments on Delta and Omicron variants
allows the estimation of the infection spreading speed and viral load. Different
variants of the SARS-CoV-2 infection are compared confirming that Omicron
is more infectious and has less severe symptoms than Delta variant. Within the
same variant, spreading speed (symptoms) correlates with viral load allowing
prognosis of disease progression.

reaction-diffusion equations,spreading speed,SARS-CoV-2 variants,viral infection,viral load

The existence and properties of intracellular waves of increased free cytoplasmic calcium concentration (calcium waves) are strongly affected by the binding and unbinding of calcium ions to a multitude of different buffers in the cell. These buffers can be mobile or immobile and, in general, have multiple binding sites that are not independent. Previous theoretical studies have focused on the case when each buffer molecule binds a single calcium ion. In this study, we analyze how calcium waves are affected by calcium buffers with two non-independent binding sites, and show that the interactions between the calcium binding sites can result in the emergence of new behaviors. In particular, for certain combinations of kinetic parameters, the profiles of buffer molecules with one calcium ion bound can be non-monotone.

Reaction-diffusion systems, Buffered calcium systems

Previous work showed that Gal-1A and Gal-8, two proteins belonging to the galactoside-binding galectin family, are the earliest determinants of the patterning of the skeletal elements of embryonic chicken limbs, and further, that their experimentally determined interactions in the embryonic limb bud can be interpreted via a reaction–diffusion–adhesion (2GL: two galectin plus ligands) model. Here, we use an ordinary differential equation-based approach to analyze the intrinsic switching modality of the 2GL network and characterize the network behavior independent of the diffusive and adhesive arms of the patterning mechanism. We identify two states: where the concentrations of both the galectins are respectively, negligible, and very high. This bistable switch-like system arises via a saddle–node bifurcation from a monostable state. For the case of mass-action production terms, we provide an explicit Lyapunov function for the system, which shows that it has no periodic solutions. Our model therefore predicts that the galectin network may exist in low expression and high expression states separated in space or time, without any intermediate states. We test these predictions in experiments performed with high density cultures of chick limb mesenchymal cells and observe that cells inside precartilage protocondensations express Gal-1A at a much higher rate than those outside, for which it was negligible. The Gal-1A and -8-based patterning network is therefore sufficient to partition the mesenchymal cell population into two discrete cell states with different developmental (chondrogenic vs. non-chondrogenic) fates. When incorporated into an adhesion and diffusion-enabled framework this system can generate a spatially patterned limb skeleton.

Limb development,Galectins,Cell state transition,Switch-like regulatory network,Ordinary differential equations,Lyapunov function,Saddle–node bifurcation

Infection spreading in cell culture occurs due to virus replication in infected cells and its random motion in the extracellular space. Multiplicity of infection experiments in cell cultures are conventionally used for the characterization of viral infection by the number of viral plaques and the rate of their growth. We describe this process with a delay reaction-diffusion system of equations for the concentrations of uninfected cells, infected cells, virus, and interferon. Time delay corresponds to the duration of viral replication inside infected cells. We show that infection propagates in cell culture as a reaction-diffusion wave, we determine the wave speed and prove its existence. Next, we carry out numerical simulations and identify three stages of infection progression: infection decay during time delay due to virus replication, explosive growth of viral load when infected cells begin to reproduce it, and finally, wave-like infection progression in cell culture characterized by a constant or slowly
growing total viral load. The modelling results are in agreement with the experimental data for the coronavirus infection in a culture of epithelial cells and for some other experiments. The presence of interferon produced by infected cells decreases the viral load but does not change the speed of infection progression in cell culture. In the 2D modelling, the total viral load grows faster than in the 1D case due to the increase of plaque perimeter.

Viral infection,cell culture,reaction-diffusion equations,time delay

We consider a class of biological models represented by a system composed of reactiondiffusion PDE coupled with difference equations (renewal equations) in n-dimensional space, with nonlocal dispersal terms and implicit time delays. The difference equation generally arises, by means of the method of characteristics, from an age-structured partial differential system. Using upper and lower solutions, we study the existence of monotonic planar traveling wave fronts connecting the extinction state to the uniform positive state. The corresponding minimum wave speed is also obtained. In addition, we investigate the effect of the parameters on this minimum wave speed and we give a detailed analysis of its asymptotic behavior

Planar monotone traveling wave front ,Reaction-diffusion PDE with delay,Difference equation,Monostable equation

A new model of viral infection spreading in cell cultures is proposed taking into account virus mutation. This model represents a reaction-diffusion system of equations with time delay for the concentrations of uninfected cells, infected cells and viral load. Infection progression is characterized by the virus replication number Rv, which determines the total viral load. Analytical formulas for the speed of propagation and for the viral load are obtained and confirmed by numerical simulations. It is shown that virus mutation leads to the emergence of a new virus variant. Conditions of the coexistence of the two variants or competitive exclusion of one of them are found, and different stages of infection progression are identified.

viral infection, mutation, cell culture, reaction-diffusion equations, time delay

The quadratic autocatalytic reaction forms a key step in a number of chemical reaction systems, and traveling waves are observed in such systems. In this study, we investigate the effect of complexation reactions on traveling waves in the quadratic autocatalytic reaction system. More precisely, under the assumption that the complexation reaction is fast relative to the autocatalytic reaction, we show that the governing system is reduced to a two-component reaction-diffusion system with density-dependent diffusivity. Further, the numerical evidence suggests that for some parameter values, a traveling wave solution of this reduced two-component system is nonlinearly selected. This is contrast to that associated with the quadratic autocatalytic reaction (without complexation reactions).

quadratic autocatalysis, complexing agent, fast reaction, traveling waves

Traveling waves form a basic class of solutions to reaction–diffusion equations, which can describe a large number of phenomena. The basic property characterizing traveling wave solutions is their speed of propagation. In this study we analyze its dependence on the diffusivities of the interacting agents. We show that this dependence is subject to some relations, which can be derived by simple scaling properties. We augment our findings by an investigation of reaction–diffusion systems describing intercellular calcium dynamics in the presence of buffer molecules. We establish some mathematical results concerning the behavior of the velocity of traveling waves in the case of fast buffer kinetics, (paying special attention to the vicinity of zero speed), and present outcomes of numerical simulations showing how complicated the interplay between the diffusion coefficients of calcium and buffering molecules can be, especially in models with more than one kind of buffer molecules.

reaction-diffusion equations, traveling waves, Euler’s homogeneous function theorem, buffered calcium systems

We analyze heteroclinic traveling waves propagating on two dimensional manifolds to show that the geometric modification of the front velocity is proportional to the geodesic curvature of the frontline. As a result, on surfaces of concave domains, stable standing fronts can be formed on lines of constant geodesic curvature. These lines minimize the geometric functional describing the system's energy, consisting of terms proportional to the front line-length and to the inclosed surface area. Front stabilization at portions of surface with negative Gaussian curvature, provides a mechanismof pattern formation. In contrast to the mechanism associated with the Turing instability, the proposed mechanism requires only a single scalar bistable reaction–diffusion equation and connects the intrinsic surface geometry with the arising pattern. By considering a system of equations modeling boundary-volume interactions, we show that polarization of the boundary may induce a corresponding polarization in the volume.

heteroclinic traveling waves, standing fronts, geodesic curvature, negative Gaussian curvature, domain polarization, pattern formation

The phenomenon of chondrogenic pattern formation in the vertebrate limb is one of the best studied examples of organogenesis. Many different models, mathematical as well as conceptual, have been proposed for it in the last fifty years or so. In this review, we give a brief overview of the fundamental biological background, then describe in detail several models which aim to describe qualitatively and quantitatively the corresponding biological phenomena. We concentrate on several new models that have been proposed in recent years, taking into account recent experimental progress. The major mathematical tools in these approaches are ordinary and partial differential equations. Moreover, we discuss models with non-local flux terms used to account for cell-cell adhesion forces and a structured population model with diffusion. We also include a detailed list of gene products and potential morphogens which have been identified to play a role in the process of limb formation and its growth.

mathematical models of chondrogenesis, reaction-diffusion equations, pattern formation, limb development

In this paper, we study propagation phenomena on the sphere using the bistable reaction–diffusion formulation. This study is motivated by the propagation of waves of calcium concentrations observed on the surface of oocytes, and the propagation of waves of kinase concentrations on the B-cell membrane in the immune system. To this end, we first study the existence and uniqueness of mild solutions for a parabolic initial-boundary value problem on the sphere with discontinuous bistable nonlinearities. Due to the discontinuous nature of reaction kinetics, the standard theories cannot be applied to the underlying equation to obtain the existence of solutions. To overcome this difficulty, we give uniform estimates on the Legendre coefficients of the composition function of the reaction kinetics function and the solution, and a priori estimates on the solution, and then, through the iteration scheme, we can deduce the existence and related properties of solutions. In particular, we prove that the constructed solutions are of C2,1 class everywhere away from the discontinuity point of the reaction term. Next, we apply this existence result to study the propagation phenomenon on the sphere. Specifically, we use stationary solutions and their variants to construct a pair of time-dependent super/sub-solutions with different moving speeds. When applied to the case of sufficiently small diffusivity, this allows us to infer that if the initial concentration of the species is above the inhomogeneous steady state, then the species will exhibit the propagating behavior.

Propagation, sphere, bistable kinetics

The formation and development of blood cells is a very complex process, called hematopoiesis. This process involves a small population of cells called hematopoietic stem cells (HSCs). The HSCs are undifferentiated cells, located in the bone marrow before they become mature blood cells and enter the blood stream. They have a unique ability to produce either similar cells (self-renewal), or cells engaged in one of different lineages of blood cells: red blood cells, white cells and platelets (differentiation). The HSCs can be either in a proliferating or in a quiescent phase. In this paper, we distinguish between dividing cells that enter directly to the quiescent phase and dividing cells that return to the proliferating phase to divide again. We propose a mathematical model describing the dynamics of HSC population, taking into account their spatial distribution. The resulting model is a coupled reaction–diffusion equation and difference equation with delay. We study the existence of monotone traveling wave fronts and the asymptotic speed of spread.

hematopoiesis, age-structured population, reaction–diffusion system with delay, difference equation, traveling wave front, asymptotic speed of spread

We prove an averaging principle which asserts convergence of diffusion processes on domains separated by semi-permeable membranes, when diffusion coefficients tend to infinity while the flux through the membranes remains constant. In the limit, points in each domain are lumped into a single state of a limit Markov chain. The limit chain's intensities are proportional to the membranes' permeability and inversely proportional to the domains' sizes. Analytically, the limit is an example of a singular perturbation in which boundary and transmission conditions play a crucial role. This averaging principle is strongly motivated by recent signaling pathways models of mathematical biology, which are discussed toward the end of the paper.

Convergence of sectorial forms and of semigroups of operators, diffusion processes, boundary and transmission conditions, Freidlin–Wentzell averaging principle, singular perturbations, signaling pathways, kinase activity, intracellular calcium dynamics, neurotransmitters

Pattern formation is one of the most fundamental yet puzzling phenomena in physics and biology. We propose that traveling front pinning into concave portions of the boundary of 3-dimensional domains can serve as a generic gradient-maintaining mechanism. Such a mechanism of domain polarization arises even for scalar bistable reaction-diffusion equations, and, depending on geometry, a number of stationary fronts may be formed leading to complex spatial patterns. The main advantage of the pinning mechanism, with respect to the Turing bifurcation, is that it allows for maintaining gradients in the specific regions of the domain. By linking the instant domain shape with the spatial pattern, the mechanism can be responsible for cellular polarization and differentiation.

The NF-κB pathway is known to transmit merely 1 bit of information about stimulus level. We combined experimentation with mathematical modeling to elucidate how information about TNF concentration is turned into a binary decision. Using Kolmogorov-Smirnov distance, we quantified the cell's ability to discern 8 TNF concentrations at each step of the NF-κB pathway, to find that input discernibility decreases as signal propagates along the pathway. Discernibility of low TNF concentrations is restricted by noise at the TNF receptor level, whereas discernibility of high TNF concentrations it is restricted by saturation/depletion of downstream signaling components. Consequently, signal discernibility is highest between 0.03 and 1 ng/ml TNF. Simultaneous exposure to TNF or LPS and a translation inhibitor, cycloheximide, leads to prolonged NF-κB activation and a marked increase of transcript levels of NF-κB inhibitors, IκBα and A20. The impact of cycloheximide becomes apparent after the first peak of nuclear NF-κB translocation, meaning that the NF-κB network not only relays 1 bit of information to coordinate the all-or-nothing expression of early genes, but also over a longer time course integrates information about other stimuli. The NF-κB system should be thus perceived as a feedback-controlled decision-making module rather than a simple information transmission channel.

cellular signaling networks, innate immunity, stress signaling

In the paper, we study some 'a priori' properties of mild solutions to a single reaction–diffusion equation with discontinuous nonlinear reaction term on the two-dimensional sphere close to its poles. This equation is the counterpart of the well-studied bistable reaction–diffusion equation on the Euclidean plane. The investigation of this equation on the sphere is mainly motivated by the phenomenon of the fertilization of oocytes or recent studies of wave propagation in a model of immune cells activation, in which the cell is modeled by a ball. Because of the discontinuous nature of reaction kinetics, the standard theory cannot guarantee the solution existence and its smoothness properties. Moreover, the singular nature of the diffusion operator near the north/south poles makes the analysis more involved. Unlike the case in the Euclidean plane, the (axially symmetric) Green's function for the heat operator on the sphere can only be represented by an infinite series of the Legendre polynomials. Our approach is to consider a formal series in Legendre polynomials obtained by assuming that the mild solution exists. We show that the solution to the equation subject to the Neumann boundary condition is C1 smooth in the spatial variable up to the north/south poles and Hölder continuous with respect to the time variable. Our results provide also a sort of 'a priori' estimates, which can be used in the existence proofs of mild solutions, for example, by means of the iterative methods.

discontinuous reaction term, stationary fronts, sphere

In the paper we examine solutions to a model of cell movement governed by the chemotaxis phenomenon derived in [14] and established via macroscopic limits of corresponding microscopic cell-based models with extended cell representations. The model is given by two PDEs for the density of cells and the concentration of a chemical. To avoid singularities in cell density, the aggregating force of chemotaxis phenomenon is attenuated by a density dependent diffusion of cells, which grows to infinity with density tending to a certain critical value. In this paper we recover the quasi-periodic structures provided by this model by means of (local in time) expansion of the solution into a basis of eigenfunctions of the linearized system. Both planar and spherical geometries are considered.

pattern formation, chemotaxis, Turing bifurcation, eigenfunction expansion

In this paper, we investigate stationary waves on the sphere using the bistable reaction-diffusion system. The motivation of this study arises from the study of activation waves of B cells in immune systems. We analytically establish (i) the existence and uniqueness of stationary waves; (ii) the limiting wave profile for small diffusivity of diffusing species; and (iii) the stability of the constructed stationary waves. The stability result may suggest the critical role of stationary waves in the determination of initial data for initiating propagating waves on the sphere, which is consistent with the numerical results for the B-cell activation model.

stationary wave, sphere, bistable kinetics

We perform a mathematical analysis of a spatially extended model describing mutual phosphorylation of cytosolic kinases and membrane receptors. The analyzed regulatory system is a part of signal transduction mechanisms, which enables communication of the cell with its extracellular environment or other cells. The mutual receptor-kinase interaction is characteristic for immune receptors and Src family kinases. From the mathematical viewpoint, the considered system is interesting because it couples differential equations defined in a domain $\Omega$ and on its boundary $\partial\Omega$ via nonlinear Robin boundary conditions. Assuming a spherically symmetric framework, our approach is to consider an auxiliary problem in which the Robin boundary condition on the external boundary of the spherical shell $\Omega$ is replaced by a uniform Dirichlet boundary condition. This method allows us to find the stationary spherically symmetric solutions, both stable and unstable. Interestingly, numerical computations suggest also the existence of nonspherically symmetric unstable stationary solutions to the spherically symmetric problem. These conjectured solutions appear to lie between super- and subsolutions that converge in time to two different stable spherically symmetric steady states. The study is completed by the proof of an existence theorem for a general version of the original system encompassing the earlier models. The theorem holds in domains of arbitrary shape, with a number of holes that may represent various organelles impenetrable to the considered kinases. The last result ensures that the problem in which the flux of active kinases (which replaces the source term) is determined by the Robin-type boundary condition at the cell membrane is well posed. The considered generalized model may thus serve as a template for intracellular signal transduction analysis.

reaction-diffusion systems, Robin-type boundary conditions, bistability, molecular signal transduction

Mitochondria not only govern energy production, but are also involved in crucial cellular signalling processes. They are one of the most important organelles determining the Ca2+ regulatory pathway in the cell. Several mathematical models explaining these mechanisms were constructed, but only few of them describe interplay between calcium concentrations in endoplasmic reticulum (ER), cytoplasm and mitochondria. Experiments measuring calcium concentrations in mitochondria and ER suggested the existence of cytosolic microdomains with locally elevated calcium concentration in the nearest vicinity of the outer mitochondrial membrane. These intermediate physical connections between ER and mitochondria are called MAM (mitochondria-associated ER membrane) complexes. We propose a model with a direct calcium flow from ER to mitochondria, which may be justified by the existence of MAMs, and perform detailed numerical analysis of the effect of this flow on the type and shape of calcium oscillations. The model is partially based on the Marhl et al model. We have numerically found that the stable oscillations exist for a considerable set of parameter values. However, for some parameter sets the oscillations disappear and the trajectories of the model tend to a steady state with very high calcium level in mitochondria. This can be interpreted as an early step in an apoptotic pathway.

As follows from experiments, waves of calcium concentration in biological tissues can be easily excited by a local mechanical stimulation. Therefore the complete theory of calcium waves should also take into account coupling between mechanical and chemical processes. In this paper we consider the existence of travelling waves for buffered systems, as in [22], completed, however, by an equation for mechanical equilibrium and respective mechanochemical coupling terms. Thus the considered, coupled system consists of reaction-diffusion equations (for the calcium and buffers concentrations) and equations for the balance of mechanical forces.

Calcium waves, reaction-diffusion systems, mechanochemical coupling

Atherosclerosis begins as an inflammation in blood vessel walls (intima). The inflammatory response of the organism leads to the recruitment of monocytes. Trapped in the intima, they differentiate into macrophages and foam cells leading to the production of inflammatory cytokines and further recruitment of white blood cells. This self-accelerating process, strongly influenced by low-density lipoproteins (cholesterol), results in a dramatic increase of the width of blood vessel walls, formation of an atherosclerotic plaque and, possibly, of its rupture. We suggest a 2D mathematical model of the initiation and development of atherosclerosis which takes into account the concentration of blood cells inside the intima and of pro- and anti-inflammatory cytokines. The model represents a reaction–diffusion system in a strip with nonlinear boundary conditions which describe the recruitment of monocytes as a function of the concentration of inflammatory cytokines. We prove the existence of travelling waves described by this system and confirm our previous results which suggest that atherosclerosis develops as a reaction–diffusion wave. The theoretical results are confirmed by the results of numerical simulations.

Atherosclerosis, Reaction–diffusion equations, Nonlinear boundary conditions, Existence of travellingwaves, Numerical simulations

Many micro-organisms use chemotaxis for aggregation, resulting in stable patterns. In this paper, the amoeba Dictyostelium discoideum serves as a model organism for understanding the conditions for aggregation and classification of resulting patterns. To accomplish this, a 1D nonlinear diffusion equation with chemotaxis that models amoeba behavior is analyzed. A classification of the steady state solutions is presented, and a Lyapunov functional is used to determine conditions for stability of inhomogenous solutions. Changing the chemical sensitivity, production rate of the chemical attractant, or domain length can cause the system to transition from having an asymptotic steady state, to having asymptotically stable single-step solution and multi-stepped stable plateau solutions.

Aggregation, Chemotaxis, Inhomogenous stability, Lyapunov functional, Plateau solutions, Dictyostelium discoideum

Mitochondria are one of the most important organelles determining Ca2+ regulatory pathway in the cell. Recent experiments suggested the existence of cytosolic microdomains with locally elevated calcium concentration (CMDs) in the nearest vicinity of the outer mitochondrial membrane (OMM). These intermediate physical connections between endoplasmic reticulum (ER) and mitochodria are called MAM (mitochondria-associated ER membrane) complexes.
The aim of this paper is to take into account the direct calcium flow from ER to mitochondria implied by the existence of MAMs and perform detailed numerical analysis of the influence of this flow on the type and shape of calcium oscillations. Depending on the permeability of MAMs interface and ER channels, different patterns of oscillations appear (simple, bursting and chaotic). For some parameters the oscillatory pattern disappear and the system tends to a steady state with extremely high calcium level in mitochondria, which can be interpreted as a crucial point at the beginning of an apoptotic pathway.

Ca2+ signaling, mitochodrial Ca2+ transport, MAMs, three pool model, bistability, apoptosis,

We proposed a spatially extended model of early events of B cell receptors (BCR) activation, which is based on mutual kinase-receptor interactions that are characteristic for the immune receptors and the Src family kinases. These interactions lead to the positive feedback which, together with two nonlinearities resulting from the double phosphorylation of receptors and Michaelis-Menten dephosphorylation kinetics, are responsible for the system bistability. We demonstrated that B cell can be activated by a formation of a tiny cluster of receptors or displacement of the nucleus. The receptors and Src kinases are activated, first locally, in the locus of the receptor cluster or the region where the cytoplasm is the thinnest. Then the traveling wave of activation propagates until activity spreads over the whole cell membrane. In the models in which we assume that the kinases are free to diffuse in the cytoplasm, we found that the fraction of aggregated receptors, capable to initiate B cell activation decreases with the decreasing thickness of cytoplasm and decreasing kinase diffusion. When kinases are restricted to the cell membrane - which is the case for most of the Src family kinases - even a cluster consisting of a tiny fraction of total receptors becomes activatory. Interestingly, the system remains insensitive to the modest changes of total receptor level. The model provides a plausible mechanism of B cells activation due to the formation of small receptors clusters collocalized by binding of polyvalent antigens or arising during the immune synapse formation.

In the paper we consider the existence of calcium travelling waves for systems with fast buffers. We prove the convergence of the travelling waves to an asymptotic limit as the kinetic coefficients characterizing the interaction between calcium and buffers tend to infinity. To be more precise, we prove the convergence of the speeds as well as the calcium component concentration profile to the profile of the travelling wave of the reduced equation. Additionally, we take into account the effect of coupling between the mechanical and chemical processes and show the existence as well the monotonicity of the profiles of concentrations. This property guarantees their positivity.

Calcium waves, Reaction–diffusion systems, Mechanochemical coupling

B and Mast cells are activated by the aggregation of the immune receptors. Motivated by this phenomena we consider a simple spatially extended model of mutual interaction of kinases and membrane receptors. It is assumed that kinase activates membrane receptors and in turn the kinase molecules bound to the active receptors are activated by transphosphorylation. Such a type of interaction implies positive feedback and may lead to bistability. In this study we apply the Steklov eigenproblem theory to analyze the linearized model and find exact solutions in the case of non-uniformly distributed membrane receptors. This approach allows us to determine the critical value of receptor dephosphorylation rate at which cell activation (by arbitrary small perturbation of the inactive state) is possible. We found that cell sensitivity grows with decreasing kinase diffusion and increasing anisotropy of the receptor distribution. Moreover, these two effects are cooperating. We showed that the cell activity can be abruptly triggered by the formation of the receptor aggregate. Since the considered activation mechanism is not based on receptor crosslinking by polyvalent antigens, the proposed model can also explain B cell activation due to receptor aggregation following binding of monovalent antigens presented on the antigen presenting cell.

We study traveling wave solutions to a system of four non-linear partial differential equations, which arise in a tissue interaction model for skin morphogenesis. Under the assumption that the strength of attachment of the epidermis to the basal lamina is sufficiently large, we prove the existence and uniqueness (up to a translation) of traveling wave solutions connecting two stationary states of the system with the dermis and epidermis cell densities being positive. We discuss the problem of the minimal wave speed.

skin morphogenesis, heteroclinic solutions, implicit function theorem

The spatiotemporal kinetics of proteins and other substrates regulate cell fate and signaling. In this study, we consider a reaction–diffusion model of interaction of membrane receptors with a two-step kinase cascade. The receptors activate the ‘up-stream’ kinase, which may diffuse over cell volume and activate the ‘down-stream’ kinase, which is also diffusing. Both kinase species and receptors are inactivated by uniformly distributed phosphatases. The positive feedback, key to the considered dynamics, arises since the up-stream kinase activates the receptors. Such a mutual interaction is characteristic for immune cell receptors. Based on the proposed model, we demonstrated that cell sensitivity (measured as a critical value of phosphatase activity at which cell maybe activated) increases with decreasing motility of receptor-interacting kinases and with increasing polarity of receptors distribution. These two effects are cooperating, the effect of receptors localisation close to one pole of the cell grows with the decreasing kinase diffusion and vanishes in the infinite diffusion limit. As the cell sensitivity increases with decreasing diffusion of receptor-interacting kinase, the overall activity of the down-stream kinase increases with its diffusion. In conclusion, the analysis of the proposed model shows that, for the fixed substrate interaction rates, spatial distribution of the surface receptors together with the motility of intracellular kinases control cell signalling and sensitivity to extracellular signals. The increase of the cell sensitivity can be achieved by (i) localisation of receptors in a small subdomain of the cell membrane, (ii) lowering the motility of receptor-interacting kinase, (iii) increasing the motility of down-stream kinases which distribute the signal over the whole cell.

In the paper we find explicit formulae for heteroclinic travelling wave solutions in the system of equations describing the dynamics of cytosolic calcium concentration and the accompanying mechanical phenomena.

calcium waves, reaction-diffusion systems, mechanochemical coupling

In living cells proteins motilities regulate the spatiotemporal dynamics of molecular pathways. We consider here a reaction–diffusion model of mutual kinase–receptor activation showing that the strength of positive feedback is controlled by the kinase diffusion coefficient. For high diffusion, the activated kinase molecules quickly leave the vicinity of the cell membrane and cannot efficiently activate the receptors. As a result, in a broad range of parameters, the cell can be activated only if the kinase diffusion coefficient is sufficiently small. Our simple model shows that change in the motility of substrates may dramatically influence the cell responses.

Reaction–diffusion system, Signal transduction, Positive feedback, Kinase activation, Membrane receptors, Protein motility

Atherosclerosis is an inflammatory disease. The atherosclerosis process starts when low-density lipoproteins (LDLs) enter the intima of the blood vessel, where they are oxidized (ox-LDLs). The anti-inflammatory response triggers the recruitment of monocytes. Once in the intima, the monocytes are transformed into macrophages and foam cells, leading to the production of inflammatory cytokines and further recruitment of monocytes. This auto-amplified process leads to the formation of an atherosclerotic plaque and, possibly, to its rupture. In this paper we develop two mathematical models based on reaction–diffusion equations in order to explain the inflammatory process. The first model is one-dimensional: it does not consider the intima’s thickness and shows that low ox-LDL concentrations in the intima do not lead to a chronic inflammatory reaction. Intermediate ox-LDL concentrations correspond to a bistable system, which can lead to a travelling wave that can be initiated by certain conditions, such as infection or injury. High ox-LDL concentrations correspond to a monostable system, and even a small perturbation of the non-inflammatory case leads to travelling-wave propagation, which corresponds to a chronic inflammatory response. The second model we suggest is two-dimensional: it represents a reaction–diffusion system in a strip with nonlinear boundary conditions to describe the recruitment of monocytes as a function of the cytokines’ concentration. We prove the existence of travelling waves and confirm our previous results, which show that atherosclerosis develops as a reaction–diffusion wave. The results of the two models are confirmed by numerical simulations. The latter show that the two-dimensional model converges to the one-dimensional one if the thickness of the intima tends to zero.

mathematical modelling, biomathematics, partial differential equations, travelling waves, reaction–diffusion equations, atherosclerosis

Existence of global classical solutions of a class of reaction–diffusion systems with chemotactic terms is demonstrated. This class contains a system of equations derived recently as a continuous limit of the stochastic discrete cellular Potts model. This provides mathematical justification for using numerical solutions of this system for modeling cellular motion in a chemotactic field.

Reaction–diffusion systems, Nonlinear diffusion, Global existence, Continuous limit, Cellular motion, Chemotaxis

Dynamical systems in which geometrically extended model cells produce and interact with diffusible (morphogen) and nondiffusible (extracellular matrix) chemical fields have proved very useful as models for developmental processes. The embryonic vertebrate limb is an apt system for such mathematical and computational modeling since it has been the subject of hundreds of experimental studies, and its normal and variant morphologies and spatiotemporal organization of expressed genes are well known. Because of its stereotypical proximodistally generated increase in the number of parallel skeletal elements, the limb lends itself to being modeled by Turing-type systems which are capable of producing periodic, or quasiperiodic, arrangements of spot- and stripe-like elements. This chapter describes several such models, including, (i) a system of partial differential equations in which changing cell density enters into the dynamics explicitly, (ii) a model for morphogen dynamics alone, derived from the latter system in the “morphostatic limit” where cell movement relaxes on a much slower time-scale than cell differentiation, (iii) a discrete stochastic model for the simplified pattern formation that occurs when limb cells are placed in planar culture, and (iv) several hybrid models in which continuum morphogen systems interact with cells represented as energy-minimizing mesoscopic entities. Progress in devising computational methods for handling 3D, multiscale, multimodel simulations of organogenesis is discussed, as well as for simulating reaction–diffusion dynamics in domains of irregular shape.

The existence and structural stability of travelling waves of systems of the free cytosolic calcium concentration in the presence of immobile buffers are studied. The proof is carried out by passing to zero with the diffusion coefficients of buffers. Thus, its method is different from Ref. 13 where the existence is proved straightforwardly.

Calcium dynamics, travelling waves, implicit function theorem

We study the existence and properties of calcium waves in the presence of buffers. The model represents a reaction–diffusion system of equations with some diffusion coefficients equal to zero. They correspond to immobile buffers. The proof of the existence of travelling waves is carried out by passing to zero in the diffusion coefficients.

We study the existence of travelling waves of the free cytosolic calcium concentration in the presence of immobile buers. We also take into account the mechanochemical interaction. In the proof we use the a priori estimates for solutions of systems with diusing buers and pass to zero with their diusion coefficients.

The paper considers conditions sufficient for the existence of classical Cmath image solutions to a new model of chondrogenesis during the vertebrate limb formation. We assume that the diffusion coefficient of the fibronectin is positive and that the function describing the interaction between the fibronectin and cells satisfies some additional properties.

bone pattern formation, chemotaxis, invariant region method

A recently proposed mathematical model of a “core” set of cellular and molecular interactions present in the developing vertebrate limb was shown to exhibit pattern-forming instabilities and limb skeleton-like patterns under certain restrictive conditions, suggesting that it may authentically represent the underlying embryonic process (Hentschel et al., Proc. R. Soc. B 271, 1713–1722, 2004). The model, an eight-equation system of partial differential equations, incorporates the behavior of mesenchymal cells as “reactors,” both participating in the generation of morphogen patterns and changing their state and position in response to them. The full system, which has smooth solutions that exist globally in time, is nonetheless highly complex and difficult to handle analytically or numerically. According to a recent classification of developmental mechanisms (Salazar-Ciudad et al., Development 130, 2027–2037, 2003), the limb model of Hentschel et al. is “morphodynamic,” since differentiation of new cell types occurs simultaneously with cell rearrangement. This contrasts with “morphostatic” mechanisms, in which cell identity becomes established independently of cell rearrangement. Under the hypothesis that development of some vertebrate limbs employs the core mechanism in a morphostatic fashion, we derive in an analytically rigorous fashion a pair of equations representing the spatiotemporal evolution of the morphogen fields under the assumption that cell differentiation relaxes faster than the evolution of the overall cell density (i.e., the morphostatic limit of the full system). This simple reaction–diffusion system is unique in having been derived analytically from a substantially more complex system involving multiple morphogens, extracellular matrix deposition, haptotaxis, and cell translocation. We identify regions in the parameter space of the reduced system where Turing-type pattern formation is possible, which we refer to as its “Turing space.” Obtained values of the parameters are used in numerical simulations of the reduced system, using a new Galerkin finite element method, in tissue domains with nonstandard geometry. The reduced system exhibits patterns of spots and stripes like those seen in developing limbs, indicating its potential utility in hybrid continuum-discrete stochastic modeling of limb development. Lastly, we discuss the possible role in limb evolution of selection for increasingly morphostatic developmental mechanisms.

Limb development, Chondrogenesis, Mesenchymal condensation, Reaction–diffusion model

In this paper we study the existence of classical solutions to a new model of skeletal development in the vertebrate limb. The model incorporates a general term describing adhesion interaction between cells and fibronectin, an extracellular matrix molecule secreted by the cells, as well as two secreted, diffusible regulators of fibronectin production, the positively-acting differentiation factor (“activator”) TGF-β, and a negatively-acting factor (“inhibitor”). Together, these terms constitute a pattern forming system of equations. We analyze the conditions guaranteeing that smooth solutions exist globally in time. We prove that these conditions can be significantly relaxed if we add a diffusion term to the equation describing the evolution of fibronectin.

In this paper we present the foundation of a unified, object-oriented, three-dimensional biomodelling environment, which allows us to integrate multiple submodels at scales from subcellular to those of tissues and organs. Our current implementation combines a modified discrete model from statistical mechanics, the Cellular Potts Model, with a continuum reaction–diffusion model and a state automaton with well-defined conditions for cell differentiation transitions to model genetic regulation. This environment allows us to rapidly and compactly create computational models of a class of complex-developmental phenomena. To illustrate model development, we simulate a simplified version of the formation of the skeletal pattern in a growing embryonic vertebrate limb.

The stability of Turing patterns in an n-dimensional cube (0, π)n is studied, where n ≥ 2. It is shown by using a generalization of a classical result of Ermentrout concerning spots and stripes in two dimensions that under appropriate assumptions only sheet-like or nodule-like structures can be stable in an n-dimensional cube. Other patterns can also be stable in regions comprising products of lower-dimensional cubes and intervals of appropriate length. Stability results are applied to a new model of skeletal pattern formation in the vertebrate limb.

The influence of mechano-chemical coupling on calcium concentration waves is considered. The propagation of calcium waves is described by a reaction–diffusion equation with the reaction term dependent on the mechanical stress responsible for the release of calcium. Similarly the balance of mechanical forces is influenced by the calcium concentration through the so-called traction force.

Calcium waves, Reaction–diffusion systems, Mechano-chemical coupling

Existence of reaction-diffusion-convection waves in unbounded strips is proved in the case of small Rayleigh numbers. In the bistable case the wave is unique, in the monostable case they exist for all speeds greater than the minimal one. The proof uses the implicit function theorem. Its application is based on the Fredholm property, index, and solvability conditions for elliptic problems in unbounded domains.

We derive a spatially extended three compartment cell model for evolution of calcium ions concetrations.
To obtain specific form of the fluxes between the compartments, we compare it with the model proposed by
Marhl et al. (2000). We examine numerically the period and shape of oscillations as a function of diffusion
coefficients. We demonstrate a decay of the oscillations at the critical value of diffusion of free calcium ions.