Recently solutions to a simple reaction diffusion system have been discovered in which localized structures (spots) make copies of themselves. In this Letter we analyze the one-dimensional analog of this process in which replication occurs until the domain is Glled with a periodic array of spots. Vfe provide a heuristic explanation of why this replication process should occur in a broad class of systems. Time dependent solutions are developed for model systems and their analytic structures investigated.PACS numbers: 82.40.Ck, 87.40.+w Over the past three decades the study of selforganization in far-from-equilibrium systems has become a major field of scientific inquiry. Within this field, the study of chemically reacting and difFusing (RD) systems has attained the status of paradigm. Although there are many reasons for the role that RD systems play, perhaps the most compelling is their obvious relevance for biological systems.Recently, Pearson [1] has observed spot patterns in a RD system that replicate themselves until they occupy the entire domain. This observation was made during a successful attempt to reproduce the labyrinthine patterns observed in [2]. In this Letter, we will look at this model system in one dimension and derive several analytic solutions to the nonlinear partial difFerential equations, including replicating spot structures. The model[3] is given by dt =7' u -uv +A(l -u), -=b V v+uv -Bv. OV 2 2 2 dt Here u(z, t) and v(x, t) are fields representing the concentrations of two chemical species with difFering difFusion coefficients, whose ratio is 62. A and B are parameters describing a feed from an external reservoir with the fixed concentrations u = 1 and v = 0.One of the most interesting structures observed by Pearson in 2D simulations consists of localized regions of high v and low u concentration, "spots, " surrounded by regions where the concentrations are nearer to the kinetics' only fixed point: u = 1, v = 0. The spots replicate: a single spot first divides into two new spots, which then separate until another replication occurs, finally filling the entire domain. Typical asymptotic configurations in these 2D simulations depend on the parameters and consist of either chaotic states in which the spots compete for territory in a continuous process of replication and death or steady states where the spots form a hexagonal pattern.In 1D, an analogous situation is realized for small enough b, although the time asymptotic state is always static in a finite domain. Figure 1 is a space-time plot of v for this case. Related phenomena have been observed by other authors [4]. In particular, Kerner and Osipov have derived numerous results on self-organization processes in active media including an analysis of the static division of one-dimensional pulses as the system size is changed.Recently, replicating spot patterns have been observed experimentally in a RD system [5]. This occurs despite 500 4QQ -~W~~0 ( 5) h o 300 4J

Punctate releases of Ca2+, called Ca2+ sparks, originate at the regular array of t-tubules in cardiac myocytes and skeletal muscle. During Ca2+ overload sparks serve as sites for the initiation and propagation of Ca2+ waves in myocytes. Computer simulations of spark-mediated waves are performed with model release sites that reproduce the adaptive Ca2+ release observed for the ryanodine receptor. The speed of these waves is proportional to the diffusion constant of Ca2+, D, rather than D, as is true for reaction-diffusion equations in a continuous excitable medium. A simplified "fire-diffuse-fire" model that mimics the properties of Ca2+-induced Ca2+ release (CICR) from isolated sites is used to explain this saltatory mode of wave propagation. Saltatory and continuous wave propagation can be differentiated by the temperature and Ca2+ buffer dependence of wave speed.

When Ca 2؉ is released from internal stores in living cells, the resulting wave of increased concentration can travel without deformation (continuous propagation) or with burst-like behavior (saltatory propagation). We analyze the ''fire-diffuse-fire'' model in order to illuminate the differences between these two modes of propagation. We show that the Ca 2؉ release wave in immature Xenopus oocytes and cardiac myocytes is saltatory, whereas the fertilization wave in the mature oocyte is continuous.Traveling waves in living cells can vary greatly in their appearance. For example, the calcium wave in immature Xenopus frog eggs propagates as a sequence of bursts (1-4) (Fig. 1a), whereas the calcium wave that occurs during fertilization in mature Xenopus eggs appears to be continuous (5, 6) ( Fig. 1b). It is commonly believed that information is encoded in the time course of the Ca 2ϩ signal (7-12). Thus, the distinction between these two modes of propagation likely has physiological significance. Remarkably, the waves in both cell types involve the release of calcium from internal stores via mechanisms that are physically and biologically similar.We report here a simple physical model that allows us to characterize these behaviors as limiting cases. A prominent feature of this model is the existence of a new time scale in addition to the usual chemical time scales, namely the intersite diffusion time, d 2 ͞D. (Here, D is the diffusion coefficient and d is the source separation.) If intersite diffusion is rate-limiting, the wave will exhibit burst-like (or saltatory) behavior (Figs. 1a and 2a). If chemical processes are rate-limiting (i.e., slow compared to intersite diffusion), then the resulting wave will be smooth (Figs. 1b and 2b). Although the intersite diffusion time is likely to be important for other wave phenomena, our analysis of recent experiments suggests that it is an essential feature for intracellular Ca 2ϩ waves.Ca 2ϩ is stored intracellularly in the endoplasmic or sarcoplasmic reticulum at 2-3 orders of magnitude greater than its concentration in the cytosol and is released by a process referred to as Ca 2ϩ -induced Ca 2ϩ release (CICR). CICR involves Ca 2ϩ release through specialized Ca 2ϩ channels that are activated at slightly elevated levels of cytosolic Ca 2ϩ and then inactivated as the level of Ca 2ϩ rises further. After an open channel closes via inactivation, it cannot reopen for some time, during which it is in a ''refractory'' state. Calcium release provides a mechanism for wave propagation (10-17), whereas inactivation limits the amount released.High-resolution imaging of Ca 2ϩ in a variety of cell types reveals localized release events called ''puffs'' (1-4) (Fig. 1a), ''sparks'' (19), or ''quantum emission domains'' (8, 9). These events have been correlated with Ca 2ϩ release from either individual or small clusters of channels. In muscle cells, sparks are observed at the t-tubule structures in the sarcoplasmic reticulum (20), which are aligned in regular arrays with a character...

The range of action of intracellular messengers is determined by their rates of diffusion and degradation. Previous measurements in oocyte cytoplasmic extracts indicated that the Ca2+-liberating second messenger inositol trisphosphate (IP3) diffuses with a coefficient (~280 μm2 s−1) similar to that in water, corresponding to a range of action of ~25 μm. Consequently, IP3 is generally considered a ‘global’ cellular messenger. We re-examined this issue by measuring local IP3-evoked Ca2+ puffs to monitor IP3 diffusing from spot photorelease in neuroblastoma cells. Fitting these data by numerical simulations yielded a diffusion coefficient (≤10 μm2 s−1) about 30 fold slower than previously reported. We propose that diffusion of IP3 in mammalian cells is hindered by binding to immobile, functionally inactive receptors that were diluted in oocyte extracts. The predicted range of action of IP3 (<5 μm) is thus smaller than the size of typical mammalian cells, indicating that IP3 should better be considered as a local rather than global cellular messenger.

We present a new class of exact solutions for the so-called Laplacian Growth Equation describing the zero-surface-tension limit of a variety of 2D pattern formation problems. Contrary to the common belief, we prove that these solutions are free of finite-time singularities (cusps) for quite general initial conditions and may well describe real fingering instabilities. At long times the interface consists of N separated moving Saffman-Taylor fingers, with "stagnation points" in between, in agreement with numerous observations. This evolution resembles the N-soliton solution of classical integrable PDE's.

In a recent article ͓Phys. Rev. Lett. 72, 2797 ͑1994͔͒ we analyzed the phenomena of self-replicating spots in the Gray-Scott model. In this article we present those results in detail and generalize them to a class of models that derives from our heuristic explanation of spot replication.

SummaryWe determine the calcium fluxes through inositol 1,4,5-trisphosphate receptor/channels underlying calcium puffs of Xenopus laevis oocytes using a simplified version of the algorithm of Ventura et al., 2005 [1]. An analysis of 130 puffs obtained with Fluo-4 indicates that Ca 2+ release comes from a region of width ~ 450 nm, that the release duration is peaked around 18ms and that the underlying Ca 2+ currents range between 0.12 and 0.95pA. All these parameters are independent of IP 3 concentration. We explore what distributions of channels that open during a puff, N p , and what relations between current and number of open channels, I(N p ), are compatible with our findings and with the distribution of puff-to-trigger amplitude ratio reported in Rose et al, 2006 [2]. To this end, we use simple "mean field" models in which all channels open and close simultaneously. We find that the variability among clusters plays an important role in shaping the observed puff amplitude distribution and that a model for which I(N p ) ~N p for small N p and I(N p )~N p 1/α (α>1) for large N p , provides the best agreement. Simulations of more detailed models in which channels open and close stochastically show that this nonlinear behavior can be attributed to the limited time resolution of the observations and to the averaging procedure that is implicit in the mean-field models. These conclusions are also compatible with observations of ~400 puffs obtained using the dye Oregon green.

A lattice Boltzmann model for reaction-diffusion systems is developed. The method provides an efficient computational scheme for simulating a variety of problems described by the reaction-diffusion equations. Diffusion phenomena, the decay to a limit cycle, and the formation of Turing patterns are studied. The results of lattice Boltzmann calculations are compared with the lattice gas method and with theoretical predictions, showing quantitative agreement. The model is extended to include velocity convection in chemically reacting fluid flows.