ON MULTITASKING IN PARALLEL CHEMICAL PROCESSORS: EXPERIMENTAL FINDINGS

2003 ◽  
Vol 13 (02) ◽  
pp. 521-533 ◽  
Author(s):  
B. P. J. DE LACY COSTELLO ◽  
A. I. ADAMATZKY
Keyword(s):  
Voronoi Diagram ◽  
Voronoi Diagrams ◽  
Classical Problem ◽  
Physical Space ◽  
Reaction Diffusion ◽  
Cross Reactivity ◽  
Chemical System ◽  
Concentration Data ◽  
Reaction Diffusion Systems ◽  
Single Chemical

A parallel chemical processor is a thin-layer of a reagent mixture which reacts to changes in its concentration — data configuration — in a predictable way to form a stationary pattern corresponding to the concentration of the reagent — result configuration. A computation in the chemical processor is implemented via the spreading and interaction of diffusive or phase waves. We design chemical processors that solve a classical problem of computational geometry — computation of a Voronoi diagram. Namely, we study the possibility of designing a multitasking chemical processor that independently and simultaneously computes Voronoi diagrams of two different data planar sets. We define a two-tasking chemical processor as two distinct reactant–substrate couples within a reaction–diffusion processor that solve separate tasks but share the same physical space. A micro-volume of the physical space is an elementary processor of a massively parallel chemical processor, therefore two reaction–diffusion systems occupying the same space are considered to be a single chemical processor. We found that when a single reactant is on a gel layer containing either one or two substrates the same single Voronoi diagram corresponding to the original location of the reactant drops is constructed. However, when two reactants are on a gel containing two substrates and where there is extremely limited cross reactivity between the separate reactant-substrate couples then two Voronoi diagrams of the data planar points (two sets of drops of separate reactants) are constructed; the third "complementary" pattern is also constructed. The first Voronoi diagram constructed is identical at least in position to the one constructed where one reactant was with one substrate (with the same original configuration of reactant drops). After the formation of the first diagram is completed the diffusion fronts corresponding to unlike reactants cross and are only annihilated where they meet another reactant front composed of the same reactant. The result is the computation of two additional Voronoi diagrams pertaining to the spatial positions of the two sets of reactant drops. The outcomes of this experiment albeit in a simple chemical system are significant because the system constitutes the first class of a synthetic chemical parallel processor capable of at least two computations at the same time.

CHEMICAL TESSELLATIONS — RESULTS OF BINARY AND TERTIARY REACTIONS BETWEEN METAL IONS AND FERRICYANIDE OR FERROCYANIDE LOADED GELS

2010 ◽  
Vol 20 (07) ◽  
pp. 2241-2252 ◽  
Author(s):  
B. P. J. DE LACY COSTELLO ◽  
I. JAHAN ◽  
P. HAMBIDGE ◽  
K. LOCKING ◽  
D. PATEL ◽  
...  
Keyword(s):  
Metal Ions ◽  
Voronoi Diagram ◽  
Metal Salt ◽  
Voronoi Diagrams ◽  
Reaction Rates ◽  
Cross Reactivity ◽  
Diffusion Reaction ◽  
Reactive Metal ◽  
Simple Chemical ◽  
Pattern Forming

In our recent letter [de Lacy Costello et al., 2009] we described the formation of spontaneous complex tessellations of the plane constructed in simple chemical reactions between drops of metal salts and ferricyanide or ferrocyanide loaded gels. In this paper, we provide more examples of binary tessellations and extend our analysis to tessellations constructed via tertiary mixtures of reactants. We also provide a classification system which describes the tessellation based on the reactivity of the metal salt with the substrate and also the cross-reactivity of the primary products. This results in balanced tessellations where both reactants have equal reactivity or unbalanced tessellations where one reactant has a lower reactivity with the gel. The products can also be partially or fully cross reactive which gives a highly complex tessellation. The tessellations are made up of colored cells (corresponding to different metal ferricyanides or ferrocyanides) separated by bisectors of low precipitate concentration. The tessellations constructed by these reactions constitute generalized Voronoi diagrams. In the case of certain binary or tertiary combinations of reactants where the diffusion/reaction rates differ, then multiplicatively weighted crystal growth Voronoi diagrams are constructed. Where one reactant has limited or no reactivity with the gel (or the products are cross reactive) then the fronts originating from the reactive metal ions cross the fronts originating from the partially reactive metal ions. The fronts can annihilate in the formation of a second Voronoi diagram relating to the relative positions of the reactive drops. Therefore, two or more generalised or weighted Voronoi diagrams can be calculated in parallel by these simple chemical systems. However when these reactions were used to calculate an additively weighted Voronoi diagram (the reaction was initiated at different time intervals) the diagram constructed did not correspond to the theoretical calculation. We use the failure of these reactions to construct an additively weighted Voronoi diagram to prove a mechanism of substrate competition for bisector formation. These tessellations are an important class of pattern forming reactions and are useful in modeling natural pattern forming phenomena in addition to being a great resource for scientific demonstrations.

scholarly journals Sequential Voronoi Diagram Calculations using Simple Chemical Reactions

2011 ◽  
Vol 3 (3) ◽  
pp. 29-41
Author(s):  
B. P. J. de Lacy Costello ◽  
I. Jahan ◽  
A. Adamatzky
Keyword(s):  
Voronoi Diagram ◽  
Chemical Reactions ◽  
Voronoi Diagrams ◽  
Cross Reactivity ◽  
Potassium Ferricyanide ◽  
Potassium Ferrocyanide ◽  
Metal Salts ◽  
Unconventional Computing ◽  
Simple Chemical ◽  
Calculation Results

In the authors’ recent paper (de Lacy Costello et al., 2010) the authors described the formation of complex tessellations of the plane arising from the various reactions of metal salts with potassium ferricyanide and ferrocyanide loaded gels. In addition to producing colourful tessellations these reactions are naturally computing generalised Voronoi diagrams of the plane. The reactions reported previously were capable of the calculation of three distinct Voronoi diagrams of the plane. As diffusion coupled with a chemical reaction is responsible for the calculation then this is achieved in parallel. Thus an increase in the complexity of the data input does not utilise additional computational resource. Additional benefits of these chemical reactions are that a permanent record of the Voronoi diagram calculation (in the form of precipitate free bisectors) is achieved, so there is no requirement for further processing to extract the calculation results. Previously it was assumed that the permanence of the results was also a potential drawback which limited reusability. This paper presents new data which shows that sequential Voronoi diagram calculations can be performed on the same chemical substrate. This is dependent on the reactivity of the original reagent and the cross reactivity of the secondary reagent with the primary product. The authors present the results from a number of binary combinations of metal salts on both potassium ferricyanide and potassium ferrocyanide substrates. The authors observe three distinct mechanisms whereby secondary sequential Voronoi diagrams can be calculated. In most cases the result was two interpenetrating permanent Voronoi diagrams. This is interesting from the perspective of mapping the capability of unconventional computing substrates. But also in the study of natural pattern formation per se.

Pattern formation in the Drosophila embryo

1981 ◽  
Vol 295 (1078) ◽  
pp. 567-594 ◽  
Keyword(s):  
Reaction Diffusion ◽  
Positional Information ◽  
Early Embryo ◽  
Current Data ◽  
Drosophila Embryo ◽  
Chemical System ◽  
Fate Map ◽  
Reaction Diffusion Systems ◽  
Epigenetic Code ◽  
Combinatorial Code

Three plausible hypotheses about developmental commitments in the Drosophila embryo propose that: (1) a micromosaic of localized determinants in the egg trigger somatic commitments; (2) monotonic anterior-posterior and dorsal-ventral gradients in the egg specify positions by a series of threshold values; (3) sequential subdivision of the early embryo into ‘anterior’ or ‘posterior’ ‘middle’ or ‘end’, ‘dorsal’ or ‘ventral’, ‘odd’ or ‘even’ compartmental domains encodes the somatic commitment in each region in a combinatorial epigenetic code. Evidence in favour of such a combinatorial code includes its capacity to account for major features of transdetermination and for many single and coordinated homoeotic transformations. In particular, both these metaplasias often cause transformations between ectodermal tissues such as antenna and genitalia, whose anlagen lie far apart on the blastoderm fate map. This phenomenon is not naturally explained by monotonic gradient models. In contrast, not only transformation between distant regions of the fate map, but also the observed geometries of compartmental boundaries on the wing, and probable ones in the early embryo, are naturally explained by reaction-diffusion models. These systems form a discrete succession of differently shaped monotonic and nonmonotonic eigenfunction gradient patterns of the same morphogens, as the tissue containing the chemical system changes in size and shape, or in other parameters. The successive mirror symmetries in non-monotonic gradients predict that distant regions of the embryo make similar developmental commitments, and also predict specific classes of pattern mutants forming mirror symmetric structures along the embryo on a variety of length scales. Finally, reaction diffusion systems spontaneously generate transverse gradients of the underlying chemicals when more than one eigenfunction is amplified at once, and therefore specify two-dimensional positional information within domains. Although it is attractive, no feature of the combinatorial code hypothesis is verified. Current data relating to whether the sequential formation of compartmental boundaries actually reflects the commitment of the two isolated ‘polyclones’ to alternative fates, whether any genes act continuously to maintain disc commitments, and whether homoeotic mutants actually ‘switch’ disc determined states, are assessed.

THE FORMATION OF VORONOI DIAGRAMS IN CHEMICAL AND PHYSICAL SYSTEMS: EXPERIMENTAL FINDINGS AND THEORETICAL MODELS

2004 ◽  
Vol 14 (07) ◽  
pp. 2187-2210 ◽  
Author(s):  
BEN DE LACY COSTELLO ◽  
NORMAN RATCLIFFE ◽  
ANDREW ADAMATZKY ◽  
ALEXEY L. ZANIN ◽  
ANDREAS W. LIEHR ◽  
...  
Keyword(s):  
Voronoi Diagram ◽  
Voronoi Diagrams ◽  
Reaction Diffusion ◽  
Theoretical Models ◽  
Gas Discharge ◽  
General Concept ◽  
Discharge System ◽  
Output State ◽  
Chemical Systems ◽  
Self Organized

The work discusses the formation of Voronoi diagrams in spatially extended nonlinear systems taking experimental and theoretical results into account. Concerning experimental systems a number of chemical systems used previously as prototype chemical processors and a barrier gas-discharge system are investigated. Although the underlying microscopic processes are very different, both types of systems show self-organized Voronoi diagrams for suitable parameters. Indeed certain chemical systems exhibit Voronoi diagrams as an output state for two distinct sets of parameters one that corresponds to the interaction of stable forced trigger waves and the other that corresponds to the spontaneous initiation and interaction of waves due to point instabilities in the system. In the case of the chemical systems front initiation, propagation and interaction (annihilation) are the primary mechanisms for Voronoi diagram formation, in the case of the barrier gas-discharge system regions of vanishing electric field define the medial axes of the Voronoi diagram. On the basis of cellular automata models the general concept of the formation of Voronoi diagrams has been explained, and related mechanisms have been simulated. Another intuitive approach towards the understanding of self-organized Voronoi diagrams has been given on the basis of reaction–diffusion models explaining the formation of Voronoi diagrams as a result of the mutual interactions of trigger fronts. The variety of systems exhibiting Voronoi diagrams as stationary states indicates that Voronoi diagrams are a generic and natural pattern formation phenomenon.

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