The cutoff phenomenon in finite Markov chains.

Natural mixing processes modeled by Markov chains often show a sharp cutoff in their convergence to long-time behavior. This paper presents problems where the cutoff can be proved (card shuffling, the Ehrenfests' urn). It shows that chains with polynomial growth (drunkard's walk) do not show cutoffs. The best general understanding of such cutoffs (high multiplicity of second eigenvalues due to symmetry) is explored. Examples are given where the symmetry is broken but the cutoff phenomenon persists.

Regulation of product chain length by isoprenyl diphosphate synthases

An analysis of the x-ray structure of homodimeric avian farnesyl diphosphate synthase (geranyltransferase, EC 2.5.1.10) coupled with information about conserved amino acids obtained from a sequence alignment of 35 isoprenyl diphosphate synthases that synthesize farnesyl (C15), geranylgeranyl (C20), and higher chain length isoprenoid diphosphates suggested that the side chains of residues corresponding to F112 and F113 in the avian enzyme were important for determining the ultimate length of the hydrocarbon chains. This hypothesis was supported by site-directed mutagenesis to transform wild-type avian farnesyl diphosphate synthase (FPS) into synthases capable of producing geranylgeranyl diphosphate (F112A), geranylfarnesyl (C25) diphosphate (F113S), and longer chain prenyl diphosphates (F112A/F113S). An x-ray analysis of the structure of the F112A/F113S mutant in the apo state and with allylic substrates bound produced the strongest evidence that these mutations caused the observed change in product specificity by directly altering the size of the binding pocket for the growing isoprenoid chain in the active site of the enzyme. The proposed binding pocket in the apo mutant structure was increased in depth by 5.8 Å as compared with that for the wild-type enzyme. Allylic diphosphates were observed in the holo structures, bound through magnesium ions to the aspartates of the first of two conserved aspartate-rich sequences (D117–D121), with the hydrocarbon tails of all the ligands growing down the hydrophobic pocket toward the mutation site. A model was constructed to show how the growth of a long chain prenyl product may proceed by creation of a hydrophobic passageway from the FPS active site to the outside surface of the enzyme.

Equilibrium distributions of microsatellite repeat length resulting from a balance between slippage events and point mutations

We describe and test a Markov chain model of microsatellite evolution that can explain the different distributions of microsatellite lengths across different organisms and repeat motifs. Two key features of this model are the dependence of mutation rates on microsatellite length and a mutation process that includes both strand slippage and point mutation events. We compute the stationary distribution of allele lengths under this model and use it to fit DNA data for di-, tri-, and tetranucleotide repeats in humans, mice, fruit flies, and yeast. The best fit results lead to slippage rate estimates that are highest in mice, followed by humans, then yeast, and then fruit flies. Within each organism, the estimates are highest in di-, then tri-, and then tetranucleotide repeats. Our estimates are consistent with experimentally determined mutation rates from other studies. The results suggest that the different length distributions among organisms and repeat motifs can be explained by a simple difference in slippage rates and that selective constraints on length need not be imposed.

Enhancing the catalytic repertoire of nucleic acids: a systematic study of linker length and rigidity

The incorporation of potentially catalytic groups in DNA is of interest for the in vitro selection of novel deoxyribozymes. A series of 10 C5-modified analogues of 2′-deoxyuridine triphosphate have been synthesised that possess side chains of differing flexibility and bearing a primary amino or imidazole functionality. For each series of nucleotide analogues differing degrees of flexibility of the C5 side chain was achieved through the use of alkynyl, alkenyl and alkyl moieties. The imidazole function was conjugated to these C5-amino-modified nucleotides using either imidazole 4-acetic acid or imidazole 4-acrylic acid (urocanic acid). The substrate properties of the nucleotides (fully replacing dTTP) with Taq polymerase during PCR have been investigated in order to evaluate their potential applications for in vitro selection experiments. 5-(3-Aminopropynyl)dUTP and 5-(E-3-aminopropenyl)dUTP and their imidazole 4-acetic acid- and urocanic acid-modified conjugates were found to be substrates. In contrast, C5-amino-modified dUTPs with alkane or Z-alkene linkers and their corresponding conjugates were not substrates. The incorporation of these analogues during PCR has been confirmed by inhibition of restriction enzyme digestion using XbaI and by mass spectrometry of the PCR products.

A modified gambler's ruin model of polyethylene chains in the amorphous region.

Polyethylene chains in the amorphous region between two crystalline lamellae M unit apart are modeled as random walks with one-step memory on a cubic lattice between two absorbing boundaries. These walks avoid the two preceding steps, though they are not true self-avoiding walks. Systems of difference equations are introduced to calculate the statistics of the restricted random walks. They yield that the fraction of loops is (2M - 2)/(2M + 1), the fraction of ties 3/(2M + 1), the average length of loops 2M - 0.5, the average length of ties 2/3M2 + 2/3M - 4/3, the average length of walks equals 3M - 3, the variance of the loop length 16/15M3 + O(M2), the variance of the tie length 28/45M4 + O(M3), and the variance of the walk length 2M3 + O(M2).

Evolutionary consequences of food chain length in kelp forest communities.

Kelp forests are strongly influenced by macroinvertebrate grazing on fleshy macroalgae. In the North Pacific Ocean, sea otter predation on macroinvertebrates substantially reduces the intensity of herbivory on macroalgae. Temperate Australasia, in contrast, has no known predator of comparable influence. These ecological and biogeographic patterns led us to predict that (i) the intensity of herbivory should be greater in temperate Australasia than in the North Pacific Ocean; thus (ii) Australasian seaweeds have been under stronger selection to evolve chemical defenses and (iii) Australasian herbivores have been more strongly selected to tolerate these compounds. We tested these predictions first by measuring rates of algal tissue loss to herbivory at several locations in Australasian and North Pacific kelp forests. There were significant differences in grazing rates among sea otter-dominated locations in the North Pacific (0-2% day-1), Australasia (5-7% day-1), and a North Pacific location lacking sea otters (80% day-1). The expectations that chronically high rates of herbivory in Australasia have selected for high concentrations of defensive secondary metabolites (phlorotannins) in brown algae and increased tolerance of these defenses in the herbivores also were supported. Phlorotannin concentrations in kelps and fucoids from Australasia were, on average, 5-6 times higher than those in a comparable suite of North Pacific algae, confirming earlier findings. Furthermore, feeding rates of Australasian herbivores were largely unaffected by phlorotannins, regardless of the compounds' regional source. North Pacific herbivores, in contrast, were consistently deterred by phlorotannins from both Australasia and the North Pacific. These findings suggest that top-level consumers, acting through food chains of various lengths, can strongly influence the ecology and evolution of plantherbivore interactions.