Graeme Mitchison

Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids

Probablistic models are becoming increasingly important in analyzing the huge amount of data being produced by large-scale DNA-sequencing efforts such as the Human Genome Project. For example, hidden Markov models are used for analyzing biological sequences, linguistic-grammar-based probabilistic models for identifying RNA secondary structure, and probabilistic evolutionary models for inferring phylogenies of sequences from different organisms. This book gives a unified, up-to-date and self-contained account, with a Bayesian slant, of such methods, and more generally to probabilistic methods of sequence analysis. Written by an interdisciplinary team of authors, it is accessible to molecular biologists, computer scientists, and mathematicians with no formal knowledge of the other fields, and at the same time presents the state of the art in this new and important field.

Maximum Discrimination Hidden Markov Models of Sequence Consensus

We introduce a maximum discrimination method for building hidden Markov models (HMMs) of protein or nucleic acid primary sequence consensus. The method compensates for biased representation in sequence data sets, superseding the need for sequence weighting methods. Maximum discrimination HMMs are more sensitive for detecting distant sequence homologs than various other HMM methods or BLAST when tested on globin and protein kinase catalytic domain sequences.

Sparse-graph codes for quantum error correction

Sparse-graph codes appropriate for use in quantum error-correction are presented. Quantum error-correcting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparse-graph codes keep the number of quantum interactions associated with the quantum error-correction process small: a constant number per quantum bit, independent of the block length. Third, sparse-graph codes often offer great flexibility with respect to block length and rate. We believe some of the codes we present are unsurpassed by previously published quantum error-correcting codes.

Phyllotaxis and the Fibonacci Series

The principal conclusion is that Fibonacci phyllotaxis follows as a mathematical necessity from the combination of an expanding apex and a suitable spacing mechanism for positioning new leaves. I have considered an inhibitory spacing mechanism at some length, as it is a plausible candidate. However, the same treatment would apply equally well to depletion of, or competition for, a compound by developing leaves, and could no doubt accommodate other ingredients. The mathematical principles involved are clear when it is assumed that only two leaves (the contacts) position a new leaf. There is some experimental evidence for this assumption. Nonetheless, it is not a precondition for Fibonacci phyllotaxis, since a computer model shows that this pattern is generated even when many leaves contribute to inhibition at a given point. Indeed, the Fibonacci pattern seems to be a robust and stable mathematical phenomenon, a finding which goes some way to explaining its widespread occurrence throughout the plant kingdom.

Neuronal branching patterns and the economy of cortical wiring

Keeping the volume of connections in the cortex as low as possible may be an important evolutionary constraint on the design of the brain. Much as an engineer tries to arrange the components of a computer in such a way as to give efficient wiring, so the brain may have evolved a layout of neuronal types which gives an economical use of axonal ‘wiring’. One key difference between computer and brain is that connections in the brain take the form of elaborate branching structures. It is argued here that certain features of cortical mapping, such as the stripes and patches seen within cortical areas, may be adaptations which allow efficient wiring by such structures. Some simple calculations are given to support this, using as models for axonal arbors certain branching patterns which give a low volume of wiring. In particular, it is shown that a pattern of stripes can give economical wiring when axon diameters follow a law dp = dp + dwith p >4, where d1 and d2 are the diameters of the daughter b

A Model for Vein Formation in Higher Plants

Experiments on vein regeneration (Jost 1942; Jacobs 1952) suggest that a signal of some kind, which can cause the differentiation of veins, flows from a source in the growing tissues of leaves towards the root. Sachs (1969, 1978) has given evidence that the capacity of a given pathway to transport this signal increases with the flux it carries. He suggests that this progress could cause the canalization of signal flow into a pattern of discrete strands, which subsequently differentiate into veins. I formulate here a mathematical model based on these assumptions, and show that it can generate well defined strands. Given plausible estimates for diffusion constants and polar transport rates, it appears that vein formation could occur by this mechanism over an appropriate distance within an acceptably short time. I show that this model can simulate Sachs’s experiments on vein formation. I also show that, with suitable assumptions about the distribution of source activity, the model can generate elaborate networks, with branches and loops of the kind seen in the leaves of higher plants.

Finding minimum entropy codes

To determine whether a particular sensory event is a reliable predictor of reward or punishment it is necessary to know the prior probability of that event. If the variables of a sensory representation normally occur independently of each other, then it is possible to derive the prior probability of any logical function of the variables from the prior probabilities of the individual variables, without any additional knowledge; hence such a representation enormously enlarges the scope of definable events that can be searched for reliable predictors. Finding a Minimum Entropy Code is a possible method of forming such a representation, and methods for doing this are explored in this paper. The main results are (1) to show how to find such a code when the probabilities of the input states form a geometric progression, as is shown to be nearly true for keyboard characters in normal text; (2) to show how a Minimum Entropy Code can be approximated by repeatedly recoding pairs, triples, etc. of an original 7-bit code for keyboard characters; (3) to prove that in some cases enlarging the capacity of the output channel can lower the entropy.

Optimal numberings of an N N array

Given a numbering of the vertices of a graph, one can define the edgesum [6] as the sum of differences between adjacent vertices. The problem of finding numberings which are optimal in the sense of minimizing the edgesum is NP-complete [2] but has been solved in the special case where the graph is the

Is there a phylogenetic signal in prokaryote proteins

2^n

The Dynamics of Auxin Transport

cube [3] and for several instances of graphs with high degrees of symmetry [6]. We find the solutions for numberings of an