Reaction-subdiffusion model of morphogen gradient formation

Short-time expansion of one-dimensional Fokker-Planck equations with heterogeneous diffusion

We formulate a short-time expansion for one-dimensional Fokker-Planck equations with spatially dependent diffusion coefficients, derived from stochastic processes with Gaussian white noise, for general values of the discretization parameter 0≤α≤1 of the stochastic integral. The kernel of the Fokker-Planck equation (the propagator) can be expressed as a product of a singular and a regular term. While the singular term can be given in closed form, the regular term can be computed from a Taylor expansion whose coefficients obey simple ordinary differential equations. We illustrate the application of our approach with examples taken from statistical physics and biophysics. Furthermore, we show how our formalism allows us to define a class of stochastic equations which can be treated exactly. The convergence of the expansion cannot be guaranteed independently from the discretization parameter α.

Effects of mortality on stochastic search processes with resetting

We study the first-passage time to the origin of a mortal Brownian particle, with mortality rate μ, diffusing in one dimension. The particle starts its motion from x>0 and it is subject to stochastic resetting with constant rate r. We first unveil the relation between the probability of reaching the target and the mean first-passage time of the corresponding problem in absence of mortality, which allows us to deduce under which conditions the former can be increased by adjusting the restart rate. We then consider the first-passage time conditioned on the event that the particle reaches the target before dying, and provide exact expressions for the mean and the variance as functions of r, corroborated by numerical simulations. By studying the impact of resetting for different mortality regimes, we also show that, if the average lifetime τ_{μ}=1/μ is long enough with respect to the diffusive time scale τ_{D}=x^{2}/(4D), there exist both a resetting rate r_{μ}^{*} that maximizes the probability and a rate r_{m} that minimizes the mean first-passage time. However, the two never coincide for positive μ, making the optimization problem highly nontrivial.

Analytical and Numerical Treatment of Continuous Ageing in the Voter Model

The conventional voter model is modified so that an agent's switching rate depends on the 'age' of the agent-that is, the time since the agent last switched opinion. In contrast to previous work, age is continuous in the present model. We show how the resulting individual-based system with non-Markovian dynamics and concentration-dependent rates can be handled both computationally and analytically. The thinning algorithm of Lewis and Shedler can be modified in order to provide an efficient simulation method. Analytically, we demonstrate how the asymptotic approach to an absorbing state (consensus) can be deduced. We discuss three special cases of the age-dependent switching rate: one in which the concentration of voters can be approximated by a fractional differential equation, another for which the approach to consensus is exponential in time, and a third case in which the system reaches a frozen state instead of consensus. Finally, we include the effects of a spontaneous change of opinion, i.e., we study a noisy voter model with continuous ageing. We demonstrate that this can give rise to a continuous transition between coexistence and consensus phases. We also show how the stationary probability distribution can be approximated, despite the fact that the system cannot be described by a conventional master equation.

Variable-order fractional master equation and clustering of particles: non-uniform lysosome distribution

In this paper, we formulate the space-dependent variable-order fractional master equation to model clustering of particles, organelles, inside living cells. We find its solution in the long-time limit describing non-uniform distribution due to a space-dependent fractional exponent. In the continuous space limit, the solution of this fractional master equation is found to be exactly the same as the space-dependent variable-order fractional diffusion equation. In addition, we show that the clustering of lysosomes, an essential organelle for healthy functioning of mammalian cells, exhibit space-dependent fractional exponents. Furthermore, we demonstrate that the non-uniform distribution of lysosomes in living cells is accurately described by the asymptotic solution of the space-dependent variable-order fractional master equation. Finally, Monte Carlo simulations of the fractional master equation validate our analytical solution. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Contact-mediated cellular communication supplements positional information to regulate spatial patterning during development

Development in multicellular organisms is marked by a high degree of spatial organization of the cells attaining distinct fates in the embryo. Recent experiments showing that suppression of intercellular interactions can alter the spatial patterns arising during development suggest that cell fates cannot be determined by the exclusive regulation of differential gene expression by morphogen gradients (the conventional view encapsulated in the French flag model). Using a mathematical model that describes the receptor-ligand interaction between cells in close physical proximity, we show that such intercellular signaling can regulate the process of selective gene expression within each cell, allowing information from the cellular neighborhood to influence the process by which the thresholds of morphogen concentration that dictate cell fates adaptively emerge. This results in local modulations of the positional cues provided by the global field set up by the morphogen, allowing interaction-mediated self-organized pattern formation to complement boundary-organized mechanisms in the context of development.

Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains

Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a one-dimensional domain that is evolving. The model equations, which have been derived from generalized continuous time random walks, can incorporate complexities such as subdiffusive transport and inhomogeneous domain stretching and shrinking. Inhomogeneously growing domains are frequently encountered in biological phenomena involving stochastic transport, such as tumor growth and morphogen gradient formation. A method for constructing analytic expressions for short-time moments of the position of the particles is developed and moments calculated from this approach are shown to compare favorably with results from random walk simulations and numerical integration of the reaction transport equation. The results show the important role played by the initial condition. In particular, it strongly affects the time dependence of the moments in the short-time regime by introducing additional drift and diffusion terms. We also discuss how our reaction transport equation could be applied to study the spreading of a population on an evolving interface. From a more general perspective, our findings help to mitigate the scarcity of analytic results for reaction-diffusion problems in geometries displaying nonuniform growth. They are also expected to pave the way for further results, including the treatment of first-passage problems associated with encounter-controlled reactions in such domains.

Time Fractional Fisher-KPP and Fitzhugh-Nagumo Equations

A standard reaction-diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction-subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction-diffusion equations. In this paper, we formulate clear examples of reaction-subdiffusion systems, based on; equal birth and death rate dynamics, Fisher-Kolmogorov, Petrovsky and Piskunov (Fisher-KPP) equation dynamics, and Fitzhugh-Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction-diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.

A matter of time: Formation and interpretation of the Bicoid morphogen gradient

Spatially distributed signaling molecules, known as morphogens, provide spatial information during development. A host of different morphogens have now been identified, from subcellular gradients through to morphogens that act across a whole embryo. These gradients form over a wide-range of timescales, from seconds to hours, and their time windows for interpretation are also highly variable; the processes of morphogen gradient formation and interpretation are highly dynamic. The morphogen Bicoid (Bcd), present in the early Drosophila embryo, is essential for setting up the future Drosophila body segments. Due to its accessibility for both genetic perturbations and imaging, this system has provided key insights into how precise patterning can occur within a highly dynamic system. Here, we review the temporal scales of Bcd gradient formation and interpretation. In particular, we discuss the quantitative evidence for different models of Bcd gradient formation, outline the time windows for Bcd interpretation, and describe how Bcd temporally adapts its own ability to be interpreted. The utilization of temporal information in morphogen readout may provide crucial inputs to ensure precise spatial patterning, particularly in rapidly developing systems.

Stochastic fluctuations and quasipattern formation in reaction-diffusion systems with anomalous transport

Many approaches to modeling reaction-diffusion systems with anomalous transport rely on deterministic equations which ignore fluctuations arising due to finite particle numbers. Starting from an individual-based model we use a generating-functional approach to derive a Gaussian approximation for this intrinsic noise in subdiffusive systems. This results in corrections to the deterministic fractional reaction-diffusion equations. Using this analytical approach, we study the onset of noise-driven quasipatterns in reaction-subdiffusion systems. We find that subdiffusion can be conducive to the formation of both deterministic and stochastic patterns. Our analysis shows that the combination of subdiffusion and intrinsic stochasticity can reduce the threshold ratio of the effective diffusion coefficients required for pattern formation to a greater degree than either effect on its own.

Effective diffusion coefficients in reaction-diffusion systems with anomalous transport

We show that the Turing patterns in reaction systems with subdiffusion can be replicated in an effective system with Markovian cross-diffusion. The effective system has the same Turing instability as the original system and the same patterns. If particles are short lived, then the transient dynamics are captured as well. We use the cross-diffusive system to define effective diffusion coefficients for the system with anomalous transport, and we show how they can be used to efficiently describe the Turing instability. We also demonstrate that the mean-squared displacement of a suitably defined ensemble of subdiffusing particles grows linearly with time, with a diffusion coefficient which agrees with our earlier calculations. We verify these deductions by numerically integrating both the fractional reaction-diffusion equation and its normally diffusing counterpart. Our findings suggest that cross-diffusive behavior can come about as a result of anomalous transport.

On the importance of protein diffusion in biological systems: The example of the Bicoid morphogen gradient

Morphogens are proteins that form concentration gradients in embryos and developing tissues, where they act as postal codes, providing cells with positional information and allowing them to behave accordingly. Bicoid was the first discovered morphogen, and remains one of the most studied. It regulates segmentation in flies, forming a striking exponential gradient along the anterior-posterior axis of early Drosophila embryos, and activating the transcription of multiple target genes in a concentration-dependent manner. In this review, the work done by us and by others to characterize the mobility of Bicoid in D. melanogaster embryos is presented. The central role played by the diffusion of Bicoid in both the establishment of the gradient and the activation of target genes is discussed, and placed in the context of the need for these processes to be all at once rapid, precise and robust. The Bicoid system, and morphogen gradients in general, remain amongst the most amazing examples of the coexistence, often observed in living systems, of small-scale disorder and large-scale spatial order. This article is part of a Special Issue entitled: Biophysics in Canada, edited by Lewis Kay, John Baenziger, Albert Berghuis and Peter Tieleman.

Continuous-time random-walk model for anomalous diffusion in expanding media

Expanding media are typical in many different fields, e.g., in biology and cosmology. In general, a medium expansion (contraction) brings about dramatic changes in the behavior of diffusive transport properties such as the set of positional moments and the Green's function. Here, we focus on the characterization of such effects when the diffusion process is described by the continuous-time random-walk (CTRW) model. As is well known, when the medium is static this model yields anomalous diffusion for a proper choice of the probability density function (pdf) for the jump length and the waiting time, but the behavior may change drastically if a medium expansion is superimposed on the intrinsic random motion of the diffusing particle. For the case where the jump length and the waiting time pdfs are long-tailed, we derive a general bifractional diffusion equation which reduces to a normal diffusion equation in the appropriate limit. We then study some particular cases of interest, including Lévy flights and subdiffusive CTRWs. In the former case, we find an analytical exact solution for the Green's function (propagator). When the expansion is sufficiently fast, the contribution of the diffusive transport becomes irrelevant at long times and the propagator tends to a stationary profile in the comoving reference frame. In contrast, for a contracting medium a competition between the spreading effect of diffusion and the concentrating effect of contraction arises. In the specific case of a subdiffusive CTRW in an exponentially contracting medium, the latter effect prevails for sufficiently long times, and all the particles are eventually localized at a single point in physical space. This "big crunch" effect, totally absent in the case of normal diffusion, stems from inefficient particle spreading due to subdiffusion. We also derive a hierarchy of differential equations for the moments of the transport process described by the subdiffusive CTRW model in an expanding medium. From this hierarchy, the full time evolution of the second-order moment is obtained for some specific types of expansion. In the case of an exponential expansion, exact recurrence relations for the Laplace-transformed moments are obtained, whence the long-time behavior of moments of arbitrary order is subsequently inferred. Our analytical and numerical results for both Lévy flights and subdiffusive CTRWs confirm the intuitive expectation that the medium expansion hinders the mixing of diffusive particles occupying separate regions. In the case of Lévy flights, we quantify this effect by means of the so-called "Lévy horizon."

Cattaneo-type subdiffusion-reaction equation

Subdiffusion in a system in which mobile particles A can chemically react with static particles B according to the rule A+B→B is considered within a persistent random-walk model. This model, which assumes a correlation between successive steps of particles, provides hyperbolic Cattaneo normal diffusion or fractional subdiffusion equations. Starting with the difference equation, which describes a persistent random walk in a system with chemical reactions, using the generating function method and the continuous-time random-walk formalism, we will derive the Cattaneo-type subdiffusion differential equation with fractional time derivatives in which the chemical reactions mentioned above are taken into account. We will also find its solution over a long time limit. Based on the obtained results, we will find the Cattaneo-type subdiffusion-reaction equation in the case in which mobile particles of species A and B can chemically react according to a more complicated rule.

Anomalous versus slowed-down Brownian diffusion in the ligand-binding equilibrium

Measurements of protein motion in living cells and membranes consistently report transient anomalous diffusion (subdiffusion) that converges back to a Brownian motion with reduced diffusion coefficient at long times after the anomalous diffusion regime. Therefore, slowed-down Brownian motion could be considered the macroscopic limit of transient anomalous diffusion. On the other hand, membranes are also heterogeneous media in which Brownian motion may be locally slowed down due to variations in lipid composition. Here, we investigate whether both situations lead to a similar behavior for the reversible ligand-binding reaction in two dimensions. We compare the (long-time) equilibrium properties obtained with transient anomalous diffusion due to obstacle hindrance or power-law-distributed residence times (continuous-time random walks) to those obtained with space-dependent slowed-down Brownian motion. Using theoretical arguments and Monte Carlo simulations, we show that these three scenarios have distinctive effects on the apparent affinity of the reaction. Whereas continuous-time random walks decrease the apparent affinity of the reaction, locally slowed-down Brownian motion and local hindrance by obstacles both improve it. However, only in the case of slowed-down Brownian motion is the affinity maximal when the slowdown is restricted to a subregion of the available space. Hence, even at long times (equilibrium), these processes are different and exhibit irreconcilable behaviors when the area fraction of reduced mobility changes.

Computational modeling reveals optimal strategy for kinase transport by microtubules to nerve terminals

Intracellular transport of proteins by motors along cytoskeletal filaments is crucial to the proper functioning of many eukaryotic cells. Since most proteins are synthesized at the cell body, mechanisms are required to deliver them to the growing periphery. In this article, we use computational modeling to study the strategies of protein transport in the context of JNK (c-JUN NH2-terminal kinase) transport along microtubules to the terminals of neuronal cells. One such strategy for protein transport is for the proteins of the JNK signaling cascade to bind to scaffolds, and to have the whole protein-scaffold cargo transported by kinesin motors along microtubules. We show how this strategy outperforms protein transport by diffusion alone, using metrics such as signaling rate and signal amplification. We find that there exists a range of scaffold concentrations for which JNK transport is optimal. Increase in scaffold concentration increases signaling rate and signal amplification but an excess of scaffolds results in the dilution of reactants. Similarly, there exists a range of kinesin motor speeds for which JNK transport is optimal. Signaling rate and signal amplification increases with kinesin motor speed until the speed of motor translocation becomes faster than kinase/scaffold-motor binding. Finally, we suggest experiments that can be performed to validate whether, in physiological conditions, neuronal cells do indeed adopt such an optimal strategy. Understanding cytoskeletal-assisted protein transport is crucial since axonal and cell body accumulation of organelles and proteins is a histological feature in many human neurodegenerative diseases. In this paper, we have shown that axonal transport performance changes with altered transport component concentrations and transport speeds wherein these aspects can be modulated to improve axonal efficiency and prevent or slowdown axonal deterioration.

Kinetics of receptor occupancy during morphogen gradient formation

During embryogenesis, sheets of cells are patterned by concentration profiles of morphogens, molecules that act as dose-dependent regulators of gene expression and cell differentiation. Concentration profiles of morphogens can be formed by a source-sink mechanism, whereby an extracellular protein is secreted from a localized source, diffuses through the tissue and binds to cell surface receptors. A morphogen molecule bound to its receptor can either dissociate or be internalized by the cell. The effects of morphogens on cells depend on the occupancy of surface receptors, which in turn depends on morphogen concentration. In the simplest case, the local concentrations of the morphogen and morphogen-receptor complexes monotonically increase with time from zero to their steady-state values. Here, we derive analytical expressions for the time scales which characterize the formation of the steady-state concentrations of both the diffusible morphogen molecules and morphogen-receptor complexes at a given point in the patterned tissue.

Nonlinear degradation-enhanced transport of morphogens performing subdiffusion

We study a morphogen gradient formation under nonlinear degradation and subdiffusive transport. In the long-time limit, we obtain the nonlinear effect of degradation-enhanced diffusion, resulting from the interaction of non-Markovian subdiffusive transport with a nonlinear reaction. We find the stationary profile of power-law type, which has implications for robustness, with the shape of the profile being controlled by the anomalous exponent. Far away from the source of morphogens, any changes in the rate of production are not felt.

Evanescent continuous-time random walks

We study how an evanescence process affects the number of distinct sites visited by a continuous-time random walker in one dimension. We distinguish two very different cases, namely, when evanescence can only occur concurrently with a jump, and when evanescence can occur at any time. The first is characteristic of trapping processes on a lattice, whereas the second is associated with spontaneous death processes such as radioactive decay. In both of these situations we consider three different forms of the waiting time distribution between jumps, namely, exponential, long tailed, and ultraslow.

Local accumulation times for source, diffusion, and degradation models in two and three dimensions

We analyze the transient dynamics leading to the establishment of a steady state in reaction-diffusion problems that model several important processes in cell and developmental biology and account for the diffusion and degradation of locally produced chemical species. We derive expressions for the local accumulation time, a convenient characterization of the time scale of the transient at a given location, in two- and three-dimensional systems with first-order degradation kinetics, and analyze their dependence on the model parameters. We also study the relevance of the local accumulation time as a single measure of timing for the transient and demonstrate that, while it may be sufficient for describing the local concentration dynamics far from the source, a more delicate multi-scale description of the transient is needed near a tightly localized source in two and three dimensions.

Random death process for the regularization of subdiffusive fractional equations

The description of subdiffusive transport in complex media by fractional equations with a constant anomalous exponent is not robust where the stationary distribution is concerned. The Gibbs-Boltzmann distribution is radically changed by even small spatial perturbations to the anomalous exponent [S. Fedotov and S. Falconer, Phys. Rev. E 85, 031132 (2012)]. To rectify this problem we propose the inclusion of the random death process in the random walk scheme, which is quite natural for biological applications including morphogen gradient formation. From this, we arrive at the modified fractional master equation and analyze its asymptotic behavior, both analytically and by Monte Carlo simulation. We show that this equation is structurally stable against spatial variations of the anomalous exponent. We find that the stationary flux of the particles has a Markovian form with rate functions depending on the anomalous rate functions, the death rate, and the anomalous exponent. Additionally, in the continuous limit we arrive at an advection-diffusion equation where advection and diffusion coefficients depend on both the death rate and anomalous exponent.

Survival probability of an immobile target in a sea of evanescent diffusive or subdiffusive traps: a fractional equation approach

We calculate the survival probability of an immobile target surrounded by a sea of uncorrelated diffusive or subdiffusive evanescent traps (i.e., traps that disappear in the course of their motion). Our calculation is based on a fractional reaction-subdiffusion equation derived from a continuous time random walk model of the system. Contrary to an earlier method valid only in one dimension (d=1), the equation is applicable in any Euclidean dimension d and elucidates the interplay between anomalous subdiffusive transport, the irreversible evanescence reaction, and the dimension in which both the traps and the target are embedded. Explicit results for the survival probability of the target are obtained for a density ρ(t) of traps which decays (i) exponentially and (ii) as a power law. In the former case, the target has a finite asymptotic survival probability in all integer dimensions, whereas in the latter case there are several regimes where the values of the decay exponent for ρ(t) and the anomalous diffusion exponent of the traps determine whether or not the target has a chance of eternal survival in one, two, and three dimensions.

Moving boundary problems governed by anomalous diffusion

Anomalous diffusion can be characterized by a mean-squared displacement 〈x(2)(t)〉 that is proportional to t(α) where α≠1. A class of one-dimensional moving boundary problems is investigated that involves one or more regions governed by anomalous diffusion, specifically subdiffusion (α<1). A novel numerical method is developed to handle the moving interface as well as the singular history kernel of subdiffusion. Two moving boundary problems are solved: the first involves a subdiffusion region to the one side of an interface and a classical diffusion region to the other. The interface will display non-monotone behaviour. The subdiffusion region will always initially advance until a given time, after which it will always recede. The second problem involves subdiffusion regions to both sides of an interface. The interface here also reverses direction after a given time, with the more subdiffusive region initially advancing and then receding.

Reaction-diffusion processes with nonlinear diffusion

We study reaction-diffusion processes with concentration-dependent diffusivity. First, the decay of the concentration in the single-species and two-species diffusion-controlled annihilation processes is determined. We then consider two natural inhomogeneous realizations. The two-species annihilation process is investigated in the situation when the reactants are initially separated, namely each species occupies a half space. In particular, we establish the growth law of the width of the reaction zone. The single-species annihilation process is studied in the situation when the spatially localized source drives the system toward the nonequilibrium steady state. Finally, we investigate a dissolution process with a localized source of diffusing atoms which react with the initially present immobile atoms forming immobile molecules.

Regulation of Turing patterns in a spatially extended chlorine-iodine-malonic-acid system with a local concentration-dependent diffusivity

A chlorine-iodine-malonic-acid Turing system involving a local concentration-dependent diffusivity (LCDD) has fundamental significance for physical, chemical, and biological systems with inhomogeneous medium. We investigated such a system by both numerical computation and mathematical analysis. Our research reveals that a variable local diffusivity has an evident effect on regulating the Turing patterns for different modes. An intrinsic square-root law is given by λ ∼ (c(1)+c(2)k)(1/2), which relates the pattern wavelength (λ) with the LCDD coefficient (k). This law indicates that the system pattern has the properties of an equivalent Turing pattern. The current study confirms that, for the Turing system with LCDD, the system pattern form retains the basic characteristics of a traditional Turing pattern in a wide range of LCDD coefficients.

Subdiffusive master equation with space-dependent anomalous exponent and structural instability

We derive the fractional master equation with space-dependent anomalous exponent. We analyze the asymptotic behavior of the corresponding lattice model both analytically and by Monte Carlo simulation. We show that the subdiffusive fractional equations with constant anomalous exponent μ in a bounded domain [0,L] are not structurally stable with respect to the nonhomogeneous variations of parameter μ. In particular, the Gibbs-Boltzmann distribution is no longer the stationary solution of the fractional Fokker-Planck equation whatever the space variation of the exponent might be. We analyze the random distribution of μ in space and find that in the long-time limit, the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events. We show that subdiffusive fractional equations with the nonuniform random distribution of anomalous exponent is an illustration of a "Black Swan," the low probability event of the small value of the anomalous exponent that completely dominates the long-time behavior of subdiffusive systems.

Delayed uncoupled continuous-time random walks do not provide a model for the telegraph equation

It has been alleged in several papers that the so-called delayed continuous-time random walks (DCTRWs) provide a model for the one-dimensional telegraph equation at microscopic level. This conclusion, being widespread now, is strange, since the telegraph equation describes phenomena with finite propagation speed, while the velocity of the motion of particles in the DCTRWs is infinite. In this paper we investigate the accuracy of the approximations to the DCTRWs provided by the telegraph equation. We show that the diffusion equation, being the correct limit of the DCTRWs, gives better approximations in L(2) norm to the DCTRWs than the telegraph equation. We conclude, therefore, that first, the DCTRWs do not provide any correct microscopic interpretation of the one-dimensional telegraph equation, and second, the kinetic (exact) model of the telegraph equation is different from the model based on the DCTRWs.

Microscopic approach to nonlinear reaction-diffusion: the case of morphogen gradient formation

We develop a microscopic theory for reaction-diffusion (RD) processes based on a generalization of Einstein's master equation [Ann. Phys. 17, 549 (1905)] with a reactive term and show how the mean-field formulation leads to a generalized RD equation with nonclassical solutions. For the nth-order annihilation reaction A+A+A+···+A→0, we obtain a nonlinear reaction-diffusion equation for which we discuss scaling and nonscaling formulations. We find steady states with solutions either exhibiting long-range power-law behavior showing the relative dominance of subdiffusion over reaction effects in constrained systems or, conversely, solutions that go to zero a finite distance from the source, i.e., having finite support of the concentration distribution, describing situations in which diffusion is slow and extinction is fast. Theoretical results are compared with experimental data for morphogen gradient formation.