Reverse-engineering of biochemical reaction networks from spatio-temporal correlations of fluorescence fluctuations

Recent developments of fluorescence labeling and highly advanced microscopy techniques have enabled observations of activities of biosignaling molecules in living cells. The high spatial and temporal resolutions of these video microscopy experiments allow detection of fluorescence fluctuations at the timescales approaching those of enzymatic reactions. Such fluorescence fluctuation patterns may contain information about the complex reaction-diffusion system driving the dynamics of the labeled molecule. Here, we have developed a method of identifying the reaction-diffusion system of fluorescently labeled signaling molecules in the cell, by combining spatio-temporal correlation function analysis of fluctuating fluorescent patterns, stochastic reaction-diffusion simulations, and an iterative system identification technique using a simulated annealing algorithm. In this report, we discuss the validity and usability of spatio-temporal correlation functions in characterizing the reaction-diffusion dynamics of biomolecules, and demonstrate application of our reaction-diffusion system identification method to a simple conceptual model for small GTPase activation.     

The role of a reaction-diffusion system in the initiation of primary hair follicles

A mechanism based on a reaction-diffusion system is proposed for the initiation of hair follicles in the epidermis during fetal development. It is demonstrated that initiation of primary follicles in a series of waves, within the proposed mechanism, is a consequence of the size and shape dependent properties of the reaction-diffusion system without the need for the propagation of signals through the skin. The observed trio grouping of follicles and variation of primary follicle density per unit skin area during development are also correctly predicted. An explanation, based on the reaction-diffusion system and the variation of its characteristic spatial wavelength with time during development, is suggested for the termination of both primary and secondary follicle initiation as well as follicle neogenesis. The proposed initiation mechanism is basically the same as that used to explain various spatial patterns observed in hair fibre formation (Nagorcka & Mooney, 1982).     

Traveling waves in a piecewise-linear reaction-diffusion model of excitable medium

One-dimensional autowaves (traveling waves) in excitable medium described by a piecewise-linear reaction-diffusion system have been investigated. Two main types of wave have been considered: a single pulse and a periodic sequence of pulses (wave trains). In a two-component system, oscillations are due to the second component of the reaction-diffusion system, while in a one-component system, they are caused by external periodic excitation (forcing). Using semianalytical solutions for the wave profile, the shape and velocity of autowaves have been found. It is shown that the dispersion relation for oscillating sequences of pulses has an anomalous character.     

A model for hydranth regeneration inTubularia

Pigment distribution presages hydranth regeneration in the marine hydroidTubularia. We suggest that such a distribution could result from a reaction-diffusion system. A model system based on a practical reaction scheme is studied and spatial structures found which closely resemble this pigment distribution. Finite-amplitude spatial structures in reaction-diffusion systems are considered. Whereas in one spatial dimension the final structures are normally very similar to the transient patterns which emerge from a linear analysis, it is shown that in more than one dimension this is not necessarily the case. The reasons for this are discussed.     

Diffusion versus network models as descriptions for the spread of prion diseases in the brain

In this paper we will discuss different modeling approaches for the spread of prion diseases in the brain. Firstly, we will compare reaction-diffusion models with models of epidemic diseases on networks. The solutions of the resulting reaction-diffusion equations exhibit traveling wave behavior on a one-dimensional domain, and the wave speed can be estimated. The models can be tested for diffusion-driven (Turing) instability, which could present a possible mechanism for the formation of plaques. We also show that the reaction-diffusion systems are capable of reproducing experimental data on prion spread in the mouse visual system. Secondly, we study classical epidemic models on networks, and use these models to study the influence of the network topology on the disease progression.     

Mixed-mode pattern in Doublefoot mutant mouse limb--Turing reaction-diffusion model on a growing domain during limb development

It has been suggested that the Turing reaction-diffusion model on a growing domain is applicable during limb development, but experimental evidence for this hypothesis has been lacking. In the present study, we found that in Doublefoot mutant mice, which have supernumerary digits due to overexpansion of the limb bud, thin digits exist in the proximal part of the hand or foot, which sometimes become normal abruptly at the distal part. We found that exactly the same behaviour can be reproduced by numerical simulation of the simplest possible Turing reaction-diffusion model on a growing domain. We analytically showed that this pattern is related to the saturation of activator kinetics in the model. Furthermore, we showed that a number of experimentally observed phenomena in this system can be explained within the context of a Turing reaction-diffusion model. Finally, we make some experimentally testable predictions.     

Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases

Bi-stable chemical systems are the basic building blocks for intracellular memory and cell fate decision circuits. These circuits are built from molecules, which are present at low copy numbers and are slowly diffusing in complex intracellular geometries. The stochastic reaction-diffusion kinetics of a double-negative feedback system and a MAPK phosphorylation-dephosphorylation system is analysed with Monte-Carlo simulations of the reaction-diffusion master equation. The results show the geometry of intracellular reaction compartments to be important both for the duration and the locality of biochemical memory. Rules for when the systems lose global hysteresis by spontaneous separation into spatial domains in opposite phases are formulated in terms of geometrical constraints, diffusion rates and attractor escape times. The analysis is facilitated by a new efficient algorithm for exact sampling of the Markov process corresponding to the reaction-diffusion master equation.     

Spatial patterns and travelling waves in population genetics.

We consider a reaction-diffusion equation to model a multi-allelic, single locus problem. The population can migrate in a homogeneous region and the diffusion rates depend upon the genotype. It is shown that if there is an equilibrium point with all alleles present and if this polymorphism is stable for the classical reaction system then it is also stable for the reaction-diffusion equation. Also a simplified model is used to investigate which allele will spread in the two-allele case. Alleles which are associated with large fitness and small dispersion do best.     

Experimental and theoretical considerations on oxygen supply for animal cell growth in fixed-bed reactors.

The supply of oxygen is a crucial parameter when cultivating animal cells in fixed-bed reactors because of the reaction-diffusion limitation within the porous carriers. To reduce limitation and increase productivity, the dissolved oxygen concentration was raised to above air saturation (hyperoxia) in long-term experiments using hybridoma cultures. This resulted in a threefold increase of the steady-state antibody production at high dilution rates compared to air saturated medium. A reaction-diffusion model was developed as a tool to describe the oxygen distribution in fixed-bed systems. The model corresponded well to the experimental data. It was also used to study the influence of several parameters on the performance of the fixed-bed system, such as the carrier size, the dissolved oxygen concentration, or the superficial flow velocity. By adapting the model it was shown that reaction-diffusion limitation is generally not a problem for other substrates such as glucose or glutamine.     

Applications of a model for scale-invariant pattern formation in developing systems

A fundamental problem in developmental biology concerns the proportioning of the developing tissue of a morphallactic system into different cell types in a way that is independent of the overall size of the tissue. The two main models for positional information in pattern formation, the source-sink models and the Turing reaction-diffusion models, have shortcomings that limit their applicability. In a previous paper, we described a model that can produce perfectly scale-invariant spatial patterns and analyzed some of its mathematical properties. In the present paper, we demonstrate some of the shortcomings of the standard reaction-diffusion models and discuss the applicability of our model to developmental systems.     

An Appropriate Bounded Invariant Region for a Bistable Reaction-Diffusion Model of the Caspase-3/8 Feedback Loop

The apoptotic caspase-3/8 feedback loop describes the core of the extrinsic pro-apoptotic signaling pathway, an essential part of apoptosis. Latter is a prototype of the programmed cell death, which enables organisms to remove damaged or infected cells. The reaction network of the caspase-3/8 feedback loop in a single cell is modeled by a reaction-diffusion system, which shows a bistable behavior. In this work, we present an appropriate bounded invariant region for the bistable reaction-diffusion system in order to theoretically confirm that diffusion rapidly balances the concentrations of the different caspase types. This justifies the decomposition of the dynamics into a diffusion dominated part on a very short time scale and a pure reaction driven dynamics on a large time scale.     

Bistable waves for a class of cooperative reaction-diffusion systems

In this paper, we consider a class of coupled cooperative reaction-diffusion systems, in which one population (or subpopulation) diffuses while the other is sedentary. We use the shooting method to prove the existence of the bistable travelling wave, and then obtain its global attractivity with phase shift and uniqueness (up to translation) via the dynamical system approach. The results are applied to some specific examples of reaction-diffusion population models.     

Reaction-Diffusion Patterns in Plant Tip Morphogenesis: Bifurcations on Spherical Caps

We study a chemical reaction-diffusion model (the Brusselator) for pattern formation on developing plant tips. A family of spherical cap domains is used to represent tip flattening during development. Applied to conifer embryos, we model the chemical prepatterning underlying cotyledon (“seed leaf”) formation, and demonstrate the dependence of patterns on tip flatness, radius, and precursor concentrations. Parameters for the Brusselator in spherical cap domains can be chosen to give supercritical pitchfork bifurcations of patterned solutions of the nonlinear reaction-diffusion system that correspond to the cotyledon patterns that appear on the flattening tips of conifer embryos.     

Mechanisms for stabilisation and destabilisation of systems of reaction-diffusion equations

Potential mechanisms for stabilising and destabilising the spatially uniform steady states of systems of reaction-diffusion equations are examined. In the first instance the effect of introducing small periodic perturbations of the diffusion coefficients in a general system of reaction-diffusion equations is studied. Analytical results are proved for the case where the uniform steady state is marginally stable and demonstrate that the effect on the original system of such perturbations is one of stabilisation. Numerical simulations carried out on an ecological model of Levin and Segel (1976) confirm the analysis as well as extending it to the case where the perturbations are no longer small. Spatio-temporal delay is then introduced into the model. Analytical and numerical results are presented which show that the effect of the delay is to destabilise the original system.     

Diffusion-induced morphogenesis in the development of Dictyostelium

A reaction-diffusion model for the regulation of cAMP in Dictyostelium discoideum is analyzed. As a parameter declines with starvation, the model sequentially yields pulse relaying, spiral waves, target patterns, streaming and sorting, directed locomotion, and tissue buckling, closely matching the observed morphogenetic sequence. These morphologies appear through successive bifurcations of a single reaction-diffusion system and do not require the expression of new genetic information.     

Morphogenesis in an asymmetric medium

Some key experiments of artificial production ofsitus inversus viscerum are briefly reviewed and a two-step mechanism for the explanation of the systematic asymmetric visceral arrangement in vertebrates is proposed. A two-variable reaction-diffusion system displaying a symmetry-breaking bifurcation is considered, and it is demonstrated that a slight asymmetry of the boundary conditions can give rise to a marked asymmetry in the resulting dissipative structure in both one-and three-dimensional systems. A criterion is formulated allowing classification of reaction-diffusion systems operating in a three-dimensional space with regard to their ability to incorporate slight asymmetries at the boundaries in the form of a chiral dissipative structure.     

强耦合竞争-竞争-互惠模型古典解的整体存在性

应用能量估计方法和bootstrap技巧证明了一类强耦合反应扩散方程系统在任意维空间中古典解的整体存在性,该系统是竞争种群含自扩散和交错扩散,互惠种群仅含自扩散的竞争-竞争-互惠模型．     

Diffusion, Cross-diffusion and Competitive Interaction

The cross-diffusion competition systems were introduced by Shigesada et al. [J. Theor. Biol. 79, 83–99 (1979)] to describe the population pressure by other species. In this paper, introducing the densities of the active individuals and the less active ones, we show that the cross-diffusion competition system can be approximated by the reaction-diffusion system which only includes the linear diffusion. The linearized stability around the constant equilibrium solution is also studied, which implies that the cross-diffusion induced instability can be regarded as Turing’s instability of the corresponding reaction-diffusion system.H. Ninomiya was Partially supported by Grant-in-Aid for Young Scientists (No. (B)15740076), Japan Society for the Promotion of Science.     

Heteroclinic waves of the FitzHugh-Nagumo equations

We study the existence and analyze the stability of heteroclinic wave solutions of a piecewise linear version of the FitzHugh-Nagumo system of reaction-diffusion equations.