Stability Analysis of a Hybrid Cellular Automaton Model of Cell Colony Growth

Cell colonies of bacteria, tumour cells and fungi, under nutrient limited growth conditions, exhibit complex branched growth patterns. In order to investigate this phenomenon we present a simple hybrid cellular automaton model of cell colony growth. In the model the growth of the colony is limited by a nutrient that is consumed by the cells and which inhibits cell division if it falls below a certain threshold. Using this model we have investigated how the nutrient consumption rate of the cells affects the growth dynamics of the colony. We found that for low consumption rates the colony takes on a Eden-like morphology, while for higher consumption rates the morphology of the colony is branched with a fractal geometry. These findings are in agreement with previous results, but the simplicity of the model presented here allows for a linear stability analysis of the system. By observing that the local growth of the colony is proportional to the flux of the nutrient we derive an approximate dispersion relation for the growth of the colony interface. This dispersion relation shows that the stability of the growth depends on how far the nutrient penetrates into the colony. For low nutrient consumption rates the penetration distance is large, which stabilises the growth, while for high consumption rates the penetration distance is small, which leads to unstable branched growth. When the penetration distance vanishes the dispersion relation is reduced to the one describing Laplacian growth without ultra-violet regularisation. The dispersion relation was verified by measuring how the average branch width depends on the consumption rate of the cells and shows good agreement between theory and simulations.Comment: 8 pages, 6 figure

Stratification of the orbit space in gauge theories. The role of nongeneric strata

Gauge theory is a theory with constraints and, for that reason, the space of physical states is not a manifold but a stratified space (orbifold) with singularities. The classification of strata for smooth (and generalized) connections is reviewed as well as the formulation of the physical space as the zero set of a momentum map. Several important features of nongeneric strata are discussed and new results are presented suggesting an important role for these strata as concentrators of the measure in ground state functionals and as a source of multiple structures in low-lying excitations.Comment: 22 pages Latex, 1 figur

Geometry, stochastic calculus and quantum fields in a non-commutative space-time

The algebras of non-relativistic and of classical mechanics are unstable algebraic structures. Their deformation towards stable structures leads, respectively, to relativity and to quantum mechanics. Likewise, the combined relativistic quantum mechanics algebra is also unstable. Its stabilization requires the non-commutativity of the space-time coordinates and the existence of a fundamental length constant. The new relativistic quantum mechanics algebra has important consequences on the geometry of space-time, on quantum stochastic calculus and on the construction of quantum fields. Some of these effects are studied in this paper.Comment: 36 pages Latex, 1 eps figur

Implications of flexible spacer rotational processes on the liquid crystal behavior of 4,5-dihydroisoxazole benzoate dimers

The synthesis of some novel non-symmetric liquid crystal dimers, {3-[4-(octyloxyphenyl)]-4,5-dihydroisoxazol-5-yl}alkyl 4-(decyloxy)benzoates (5a–d) and 4-{3-[4-(octyloxyphenyl)]-4,5-dihydroisoxazol-5-yl}alkyl 4-{[6-(octyloxy)naphthalen-2-yl]ethynyl}benzoate (9a–d), are reported. The liquid-crystalline properties, theoretical calculations based on the conformational aspects of the flexible alkyl spacer and X-ray experiments are discussed. The syntheses of the key intermediates, 2-{3-[4-(octyloxy)phenyl]-4,5-dihydroisoxazol-5-yl}alkanol (3a–d), presenting the flexible alkyl spacer were achieved through [3+2] cycloaddition reactions between nitrile oxides, which were generated in situ by oxidation of the respective aromatic oximes, and dipolarophile alkenols (CH2[double bond, length as m-dash]CH(CH2)nOH, n = 1, 2, 3, and 4). The benzoates 5a–d were synthesized through esterification of 3a–d and p-n-decyloxybenzoic acid (4). The esters 9a–d were synthesized through derivatization of isoxazolines 3a–d into 4-{3-[4-(octyloxyphenyl)]-4,5-dihydroisoxazol-5-yl}alkyl 4-bromobenzoate (7a–d) followed by a Sonogashira reaction with 2-ethynyl-6-octyloxynaphthalene (8). 5a and 5b showed a monotropic smectic C phase. 9a/c displayed a enantiotropic nematic (N) mesophase, whereas 9b/d showed a monotropic nematic mesophase. No mesophase was observed for 7a–d. An odd–even effect was observed for 5a–d and 9a–d associated with the crystal to isotropic phase transition and crystal to nematic phase, respectively, as the length of the spacer was increased from 1 to 4 carbon atoms. The transitional properties were higher for odd-numbered members (n = 1 and 3) for all of the series studied. The X-ray data of compounds 5a and 5b are in agreement with polarizing optical microscopy observations with the assignment of an SmC mesophase. Density functional theory calculations using the B3LYP hybrid functional with the level 6-311G(d,p) basis set were performed for molecules 5a–d to correlate the conformation of the flexible spacer and the transitional properties. The conformational analysis showed that the most stable conformation for 5a–d is one where all of the carbon atoms of the flexible spacer are orientated at 180° (antiperiplanar orientation) except for 5a because the spacer is too short. The odd-numbered members have a more bent shape and are less elongated molecules than the even-numbered members. Thus, mesomorphic behavior is dictated by the conformational constraint imposed by the flexible spacer on the mesogenic groups