Power spectra methods for a stochastic description of diffusion on deterministically growing domains

A central challenge in developmental biology is understanding the creation of robust spatiotemporal heterogeneity. Generally, the mathematical treatments of biological systems have used continuum, mean-field hypotheses for their constituent parts, which ignores any sources of intrinsic stochastic effects. In this paper we consider a stochastic space-jump process as a description of diffusion, i.e., particles are able to undergo a random walk on a discretized domain. By developing analytical Fourier methods we are able to probe this probabilistic framework, which gives us insight into the patterning potential of diffusive systems. Further, an alternative description of domain growth is introduced, with which we are able to rigorously link the mean-field and stochastic descriptions. Finally, through combining these ideas, it is shown that such stochastic descriptions of diffusion on a deterministically growing domain are able to support the nucleation of states that are far removed from the deterministic mean-field steady state

Stochastic reaction & diffusion on growing domains: understanding the breakdown of robust pattern formation

Many biological patterns, from population densities to animal coat markings, can be thought of as heterogeneous spatiotemporal distributions of mobile agents. Many mathematical models have been proposed to account for the emergence of this complexity, but, in general, they have consisted of deterministic systems of differential equations, which do not take into account the stochastic nature of population interactions. One particular, pertinent criticism of these deterministic systems is that the exhibited patterns can often be highly sensitive to changes in initial conditions, domain geometry, parameter values, etc. Due to this sensitivity, we seek to understand the effects of stochasticity and growth on paradigm biological patterning models. In this paper, we extend spatial Fourier analysis and growing domain mapping techniques to encompass stochastic Turing systems. Through this we find that the stochastic systems are able to realize much richer dynamics than their deterministic counterparts, in that patterns are able to exist outside the standard Turing parameter range. Further, it is seen that the inherent stochasticity in the reactions appears to be more important than the noise generated by growth, when considering which wave modes are excited. Finally, although growth is able to generate robust pattern sequences in the deterministic case, we see that stochastic effects destroy this mechanism for conferring robustness. However, through Fourier analysis we are able to suggest a reason behind this lack of robustness and identify possible mechanisms by which to reclaim it

Non-linear effects on Turing patterns: time oscillations and chaos.

We show that a model reaction-diffusion system with two species in a monostable regime and over a large region of parameter space, produces Turing patterns coexisting with a limit cycle which cannot be discerned from the linear analysis. As a consequence, Turing patterns oscillate in time, a phenomenon which is expected to occur only in a three morphogen system. When varying a single parameter, a series of bifurcations lead to period doubling, quasi-periodic and chaotic oscillations without modifying the underlying Turing pattern. A Ruelle-Takens-Newhouse route to chaos is identified. We also examined the Turing conditions for obtaining a diffusion driven instability and discovered that the patterns obtained are not necessarily stationary for certain values of the diffusion coefficients. All this results demonstrates the limitations of the linear analysis for reaction-diffusion systems

Effects of intrinsic stochasticity on delayed reaction-diffusion patterning systems

Cellular gene expression is a complex process involving many steps, including the transcription of DNA and translation of mRNA; hence the synthesis of proteins requires a considerable amount of time, from ten minutes to several hours. Since diffusion-driven instability has been observed to be sensitive to perturbations in kinetic delays, the application of Turing patterning mechanisms to the problem of producing spatially heterogeneous differential gene expression has been questioned. In deterministic systems a small delay in the reactions can cause a large increase in the time it takes a system to pattern. Recently, it has been observed that in undelayed systems intrinsic stochasticity can cause pattern initiation to occur earlier than in the analogous deterministic simulations. Here we are interested in adding both stochasticity and delays to Turing systems in order to assess whether stochasticity can reduce the patterning time scale in delayed Turing systems. As analytical insights to this problem are difficult to attain and often limited in their use, we focus on stochastically simulating delayed systems. We consider four different Turing systems and two different forms of delay. Our results are mixed and lead to the conclusion that, although the sensitivity to delays in the Turing mechanism is not completely removed by the addition of intrinsic noise, the effects of the delays are clearly ameliorated in certain specific cases

Convex Independence in Permutation Graphs

A set C of vertices of a graph is P_3-convex if every vertex outside C has at most one neighbor in C. The convex hull \sigma(A) of a set A is the smallest P_3-convex set that contains A. A set M is convexly independent if for every vertex x \in M, x \notin \sigma(M-x). We show that the maximal number of vertices that a convexly independent set in a permutation graph can have, can be computed in polynomial time

Random matrix ensembles with an effective extensive external charge

Recent theoretical studies of chaotic scattering have encounted ensembles of random matrices in which the eigenvalue probability density function contains a one-body factor with an exponent proportional to the number of eigenvalues. Two such ensembles have been encounted: an ensemble of unitary matrices specified by the so-called Poisson kernel, and the Laguerre ensemble of positive definite matrices. Here we consider various properties of these ensembles. Jack polynomial theory is used to prove a reproducing property of the Poisson kernel, and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the Dyson circular ensemble. For the Laguerre ensemble, the scaled global density is calculated exactly for all even values of the parameter $\beta$, while for $\beta = 2$ (random matrices with unitary symmetry), the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class.Comment: LaTeX209, 17 page

Probing for evolutionary links between local ULIRGs and QSOs from NIR spectroscopy

We present a study of the dynamical evolution of Ultraluminous Infrared Galaxies (ULIRGs), merging galaxies of infrared luminosity >10^12 L_sun. During our Very Large Telescope large program, we have obtained ISAAC near-infrared, high-resolution spectra of 54 ULIRGs (at several merger phases) and 12 local Palomar-Green QSOs to investigate whether ULIRGs go through a QSO phase during their evolution. One possible evolutionary scenario is that after nuclear coalescence, the black hole radiates close to Eddington to produce QSO luminosities. The mean stellar velocity dispersion that we measure from our spectra is similar (~160 km/s) for 30 post-coalescence ULIRGs and 7 IR-bright QSOs. The black holes in both populations have masses of order 10^7-10^8 M_sun (calculated from the relation to the host dispersion) and accrete at rates >0.5 Eddington. Placing ULIRGs and IR-bright QSOs on the fundamental plane of early-type galaxies shows that they are located on a similar region (that of moderate-mass ellipticals), in contrast to giant ellipticals and radio-loud QSOs. While this preliminary comparison of the ULIRG and QSO host kinematical properties indicates that (some) ULIRGs may undergo a QSO phase in their evolutionary history before they settle down as ellipticals, further data on non-IR excess QSOs are necessary to test this scenario.Comment: To appear in the "QSO Host Galaxies: Evolution and Environment" conference proceedings; meeting held in Leiden, August 200

Comparison of Post-injection Site Pain Between Technetium Sulfur Colloid and Technetium Tilmanocept in Breast Cancer Patients Undergoing Sentinel Lymph Node Biopsy.

BackgroundNo prior studies have examined injection pain associated with Technetium-99m Tilmanocept (TcTM).MethodsThis was a randomized, double-blinded study comparing postinjection site pain between filtered Technetium Sulfur Colloid (fTcSC) and TcTM in breast cancer lymphoscintigraphy. Pain was evaluated with a visual analogue scale (VAS) (0-100 mm) and the short-form McGill Pain Questionnaire (SF-MPQ). The primary endpoint was mean difference in VAS scores at 1-min postinjection between fTcSC and TcTM. Secondary endpoints included a comparison of SF-MPQ scores between the groups at 5 min postinjection and construction of a linear mixed effects model to evaluate the changes in pain during the 5-min postinjection period.ResultsFifty-two patients underwent injection (27-fTcSC, 25-TcTM). At 1-min postinjection, patients who received fTcSC experienced a mean change in pain of 16.8 mm (standard deviation (SD) 19.5) compared with 0.2 mm (SD 7.3) in TcTM (p = 0.0002). At 5 min postinjection, the mean total score on the SF-MPQ was 2.8 (SD 3.0) for fTcSC versus 2.1 (SD 2.5) for TcTM (p = 0.36). In the mixed effects model, injection agent (p < 0.001), time (p < 0.001) and their interaction (p < 0.001) were associated with change in pain during the 5-min postinjection period. The model found fTcSC resulted in significantly more pain of 15.2 mm (p < 0.001), 11.3 mm (p = 0.001), and 7.5 mm (p = 0.013) at 1, 2, and 3 min postinjection, respectively.ConclusionsInjection with fTcSC causes significantly more pain during the first 3 min postinjection compared with TcTM in women undergoing lymphoscintigraphy for breast cancer

Scaled free energies, power-law potentials, strain pseudospins and quasi-universality for first-order structural transitions

We consider ferroelastic first-order phase transitions with $N_{OP}$ order-parameter strains entering Landau free energies as invariant polynomials, that have $N_V$ structural-variant Landau minima. The total free energy includes (seemingly innocuous) harmonic terms, in the $n = 6 -N_{OP}$ {\it non}-order-parameter strains. Four 3D transitions are considered, tetragonal/orthorhombic, cubic/tetragonal, cubic/trigonal and cubic/orthorhombic unit-cell distortions, with respectively, $N_{OP} = 1, 2, 3$ and 2; and $N_V = 2, 3, 4$ and 6. Five 2D transitions are also considered, as simpler examples. Following Barsch and Krumhansl, we scale the free energy to absorb most material-dependent elastic coefficients into an overall prefactor, by scaling in an overall elastic energy density; a dimensionless temperature variable; and the spontaneous-strain magnitude at transition $\lambda <<1$. To leading order in $\lambda$ the scaled Landau minima become material-independent, in a kind of 'quasi-universality'. The scaled minima in $N_{OP}$-dimensional order-parameter space, fall at the centre and at the $N_V$ corners, of a transition-specific polyhedron inscribed in a sphere, whose radius is unity at transition. The `polyhedra' for the four 3D transitions are respectively, a line, a triangle, a tetrahedron, and a hexagon. We minimize the $n$ terms harmonic in the non-order-parameter strains, by substituting solutions of the 'no dislocation' St Venant compatibility constraints, and explicitly obtain powerlaw anisotropic, order-parameter interactions, for all transitions. In a reduced discrete-variable description, the competing minima of the Landau free energies induce unit-magnitude pseudospin vectors, with $N_V +1$ values, pointing to the polyhedra corners and the (zero-value) center.Comment: submitted to PR