TURES

We make the simplifying assumption that both and are functions only of temperature, not of pressure and composition. For a condensable component it follows that at the same tempera-ture, . ... 

It has become increasingly appreciated in recent years that the surface stmc-ture of the adsorbent may be altered in the adsorption process. Qualitatively,... 

Diflfiisive processes nonnally operate in chemical systems so as to disperse concentration gradients. In a paper in 1952, the mathematician Alan Turing produced a remarkable prediction [37] that if selective diffiision were coupled with chemical feedback, the opposite situation may arise, with a spontaneous development of sustained spatial distributions of species concentrations from initially unifonn systems. Turmg s paper was set in the context of the development of fonn (morphogenesis) in embryos, and has been adopted in some studies of animal coat markings. With the subsequent theoretical work at Brussels [1], it became clear that oscillatory chemical systems should provide a fertile ground for the search for experimental examples of these Turing patterns. 

Figure A3.14.12. The first experimental observation of a Turing pattern in a gel strip reactor. Solutions containing separate components of the CIMA/CDIMA reaction are flowed along each edge of the strip and a spatial pattern along the horizontal axis develops for a range of experimental conditions. (Reprinted with pennission from [38], The American Physical Society.)...

Figure A3.14.13. Further examples of the various Turing patterns observable in a 2D gel reaetor. (a) and (b) spots, (e) and (d) stripes, (e) and (1) wider field of view showing long-range defeets in basie stnietine. The seale bar alongside eaeh figure represents 1 nnu. (Reprinted with pemrission from [39], The Ameriean Institute of Physios.)...

The search for Turing patterns led to the introduction of several new types of chemical reactor for studying reaction-diffusion events in feedback systems. Coupled with huge advances in imaging and data analysis capabilities, it is now possible to make detailed quantitative measurements on complex spatiotemporal behaviour. A few of the reactor configurations of interest will be mentioned here. 

Lengyel I and Epstein I R 1992 A chemical approach to designing Turing patterns in reaction-diffusion systems Proc. Natl Acad. Sc/. 89 3977-9... 

Castets V, Dulos E, Boissonade J and De Kepper P 1990 Experimental evidence of a sustained standing Turing-type nonequilibrium structure Rhys. Rev. Lett. 64 2953-6... 

Borokmans P, Dewel G, De Wit A and Walgraef D Turing bifuroations and pattern seleotion Chemical M/aves and Patterns eds R Kapral and K Showalter (Dordreoht Kluwer) eh 10, pp 323-63... 

Now we may state tlie well known conditions for a Turing bifurcation. If A <0 and A22 < 0 we say species A j is tlie activator and speciesis tlie inliibitor. Then, for a Turing bifurcation to occur we must have detB = 0,Tt B> 0 and A 7)2 + > 0. The (unique) wavenumber at tlie bifurcation is... 

Consequently, when D /Dj exceeds the critical value, close to the bifurcation one expects to see the appearance of chemical patterns with characteristic lengtli i= In / k. Beyond the bifurcation point a band of wave numbers is unstable and the nature of the pattern selected (spots, stripes, etc.) depends on the nonlinearity and requires a more detailed analysis. Chemical Turing patterns were observed in the chlorite-iodide-malonic acid (CIMA) system in a gel reactor [M, 59 and 60]. Figure C3.6.12(a) shows an experimental CIMA Turing spot pattern [59]. 

Figure C3.6.12 a) Turing spot pattern in the CIMA reaction. (A) Tio-temporal turbulence near the Turing bifurcation. Reproduced by pennission from Ouyang and Swinney [59].

The Turing mechanism requires that the diffusion coefficients of the activator and inlribitor be sufficiently different but the diffusion coefficients of small molecules in solution differ very little. The chemical Turing patterns seen in the CIMA reaction used starch as an indicator for iodine. The starch indicator complexes with iodide which is the activator species in the reaction. As a result, the complexing reaction with the immobilized starch molecules must be accounted for in the mechanism and leads to the possibility of Turing pattern fonnation even if the diffusion coefficients of the activator and inlribitor species are the same 62. 

One may also observe a transition to a type of defect-mediated turbulence in this Turing system (see figure C3.6.12 (b). Here the defects divide the system into domains of spots and stripes. The defects move erratically and lead to a turbulent state characterized by exponential decay of correlations [59]. Turing bifurcations can interact with the Hopf bifurcations discussed above to give rise to very complicated spatio-temporal patterns [63, 64]. 

We can approximate this firaction of states in the reactant well, by expanding the potential in a harmonic approximation and assuming that the tempera ture is low compared with the barrier height. This leads to an estimate for the rate constant... 

To compute densities for such a large number of distributions, reliable and fully automatic density estimators are necessary. The only density estimators fliscussed in the protein litera,ture are histogram estimates. However, these are nonsmooth and thus not suitable for global optimization techniques that c-ombine local and global search. Moreover, for a sample of size n and an optimally chosen bin size, histogram estimates have an accuracy of This is an extremely poor accurar y, far away from the theoretically attainable accuracy of other density estimators. (To reach = Q.l one... 

These examples show quite dearly the dose relationship between this new stmc-ture representation embedded in RAMSES and MO Theory. 

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In a reaction, bonds are broken and made. In some cases free electrons are shifted also. The rcaciion center contains all the bond.s being broken or made during the reaction as well as all the electron rearrangement processes. The reaction uhstme-ture is the structural subunit of atoms and bonds around the reaction center that has to be present in a compound in order for the reaction to proceed in the foi"ward (synthesis) direction (Figure 10,3-32). Both characteristics of a reaction can be used to. search for reactions with an identical reaction center and reaction substructure but with different structural units beyond the reaction substructure. For example, this can be achieved by searching in a reaction database. 

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