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I am a physicist at the Curie Institute/CNRS and the London Institute for Mathematical Sciences; I also periodically visit Cambridge.

I use statistical mechanics to study complex systems in physics and interdisciplinary fields. My research interests are discrete dynamics, complex networks and fundamental laws of biology.

I have also written two popular books: The Man's Book, an almanac for men; and The 85 Ways to Tie a Tie, a book about ties and tie knots.

Dynamics of single input networks

self-assembly on a square lattice We study boolean dynamics on the simplest class of network topologies: those in which each node has a single input (K = 1). Despite their simplicity, they exhibit highly intricate bahaviour. We give the exact solution for the size and number of attractors on a loop and multiple loops of the same size. By expressing the dynamics of a network as a composition of the dynamics of its modules, we give a detailed solution to the critical K = 1 Kauffman model, and show that the minimum number of attractors scales as 2^(n-o(n)), where n is the number of nodes in loops.

A book to train manly men

Arnold Schwarzenegger training The third and most recent edition of The Man's Book was published by Little, Brown in the United States on 6 May, 2009. Expanded, revised and retypeset, this is the first time that the definitive almanac for men has been available in America. New features include: 16 more sections, 35 more figures and numerically indexed sections and subsections.

Self-assembly

self-assembly on a square lattice Recently I have been working on self-assembly: when you put many copies of one or more building blocks into a box and shake it, a macroscopic structure spontaneously emerges. The figure to the right shows two building blocks on the square lattice which interact along edges according to whether the edge colours match up. Self-assembly occurs in the formation of protein complexes in biology, the spread of prion and other amyloid diseases, novel materials and crystals ex nihilo, and is associated with regular and aperiodic tiling. We are interested in enumerating regular-polygonal blocks and their associated structures, as well as those of multiple species of blocks.

Directed Networks

Clustering signatures classify directed networks We use a clustering signature, based on a recently introduced generalization of the clustering coefficient to directed networks, to analyze 16 directed real-world networks of five different types: social networks, genetic transcription networks, word adjacency networks, food webs and electric circuits. We show that these five classes of networks are cleanly separated in the space of clustering signatures due to the statistical properties of their local neighbourhoods, demonstrating the usefulness of clustering signatures as a classifier of directed networks.

Non-coding DNA

noncoding DNA network How much non-coding DNA do eukaryotes require? In most eukaryotes, a large proportion of the genome does not code for proteins. The non-coding part is observed to vary greatly in size even between closely related species. We report evidence that eukaryotes require a certain minimum amount of non-coding DNA (ncDNA), and that this minimum increases quadratically with the amount of coding DNA. Based on a simple model of the growth of regulatory networks, we derive a theoretical prediction of the required quantity of ncDNA and find it to be in excellent agreement with observation.

Mathematics of Conkers

DARPA FunBio logo Conkers is a popular game played with the nuts of the common horse-chestnut tree (conkers), in which pairs of players take turns swiping each other's conker with their own until one conker is broken. Each conker is assigned a score as follows. All new conkers start with a score of 1. Each time a conker beats another conker, it adds to its score the score of the defeated.

In Single elimination competition, we study a simple model of competition in which each player has a fixed strength. Randomly selected pairs of players compete and the stronger one wins. We show that the best indicator of future success is not the number of wins but a player's wealth: the accumulated wealth of all defeated players. We calculate statistics of a conker with a given score and offer advice on strategy.

Fundamental Laws of Biology

DARPA FunBio logo In the summer of 2007 I joined the DARPA Fundamental Laws of Biology (FunBio) program. Organised by the Defense Sciences Office, it is a team of biologists, mathematicians, and physicists working together to

The Man's Book 2008

How to hold a cigarette The Man's Book is the authoritative handbook of men's customs, habits and pursuits - a vade mecum for modern-day manliness. At a time when the sexes are muddled and masculinity is marginalized, The Man's Book unabashedly celebrates being male. Chaps, cads, blokes and bounders, rejoice: The Man's Book will bring you back to wear you belong.

The 2008 edition of The Man's Book, published on 19 September, 2007, has been expanded, revised and updated. A free sample chapter can be downloaded.

Dynamics of Network Motifs

Complex dynamical networks are found in biology, technology and sociology. Recently it was observed that some local network structures, or network motifs, are much more frequently observed in complex networks than would be expected by chance. Although this biased distribution of motifs appears to apply to a broad range of networks, it remains unclear why some motifs are ubiquitous and others are not.

In a recent paper, we study the dynamics of network motifs by treating them as small Boolean networks and comprehensively evaluating all possible binary (Boolean) update rules. We introduce a formalism for classifying dynamical behaviour and show that some network motifs are fundamentally more versatile—capable of executing a variety of tasks—than others. Because it is more likely to be capable of an arbitrary task, a versatile motif would ostensibly occur more frequently in real networks, all else being equal. While versatility roughly increases with motif complexity, we find evidence of a critical complexity, after which increasing the number of bonds confers no advantage.

Can Biology Lead to New Theorems?

Fisher 7x7 Latin square

R. A. Fisher 7x7 Latin
square, from Caius
College Hall, Cambridge

More often than not, collaboration between biologists and mathematical scientists is a one-sided affair. Mathematicians and physicists are under pressure to make real biological contributions, irrespective of whether the mathematics is of interest or importance. From their point of view, such interdisciplinary collaborations are of limited real interest.

This picture is starting to change. In '...Biology Is Mathematics' Next Physics, Only Better', Joel Cohen argues that in the 21st century, biology, like physics in the century before it, will drive mathematics in new directions. Bernd Sturmfels offers more explicit evidence in his paper 'Can Biology Lead to New Theorems?'. He outlines four theorems which are direct outgrowths of biological research.

These prognostications suggest an exciting future for quantitative biology. While much of systems biology is applied (it has immediate practical applications), it is now clear that it also comprises basic science. New mathematical understanding, and new mathematics, are needed to make sense of phenomena ranging from the complex dynamics in regulatory networks to the faint deviation from randomness in genetic expression curves.

Here is an example from my work where studying microarray expression led to a theorem in number theory (Microarray expression series, Properties of up-down numbers). Let CN(s) be the number of permutations of length N+1 with a given up-down signature s. Then, for p prime,

Cp(s) = -1,0,1 (mod p)

Thomas Fink

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