Stochastic gene expression as a many-body problem

Abstract

Gene expression has a stochastic component because of the singlemolecule nature of the gene and the small number of copies of individual DNA-binding proteins in the cell. We show how the statistics of such systems can be mapped onto quantum many-body problems. The dynamics of a single gene switch resembles the spin-boson model of a two-site polaron or an electron transfer reaction. Networks of switches can be approximately described as quantum spin systems by using an appropriate variational principle. In this way, the concept of frustration for magnetic systems can be taken over into gene networks. The landscape of stable attractors depends on the degree and style of frustration, much as for neural networks. We show the number of attractors, which may represent cell types, is much smaller for appropriately designed weakly frustrated stochastic networks than for randomly connected networks.T he complexity of a cell's genome is expressed through the interactions of many genes with a large variety of proteins. Understanding gene expression, therefore, is a many-body problem. But what kind of many-body problem? Should we think of gene expression using the metaphors and techniques of deterministic many-body problems like those developed for the ''clockwork Universe'' of 19th century celestial mechanics? Or is it appropriate to use statistical ideas like those that form the language of condensed matter physics and physical chemistry (1)?The deterministic view has much to recommend it. Miracles of development require intricacy and precision (2). Cell cycles, a prominent dynamic sign of life not found in inanimate matter, are often described as clocks. With the great information content of the genome now so apparent in the ''postgenomic era,'' it is hard to resist making analogies between cells and those manmade information processors, electronic computers, which grind through their programs with a determination that Laplace would have found thrilling. The stochastic view is not without merit, however. Because a gene is a molecule, the statistical fluctuations of atomism cannot be avoided, as Delbrück realized so long ago (3). The technological capabilities of modern experimental biophysics have also made the presence of stochastic behavior in cells undeniable as an experimental fact (4). Under some circumstances, the game theoretic advantage of unpredictable behavior in predator-prey relations among single-cell organisms will be a clear incentive for stochasticity to have evolved adaptively. Furthermore, even when modern cells have well orchestrated patterns of gene expression, we need to know how this elegant patterning can have been achieved in the light of there being both specific and nonspecific interactions of DNA-binding proteins with the myriad possible similar but nevertheless incorrect sites along the genome, many of which remain silent.The main purpose of this paper is to begin the exploration of stochastic gene expression by developing an analogy to quantum many-body problems. Theoretical work on stochasti...

Nonequilibrium landscape theory of neural networks

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