We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

The reference human genome sequence set the stage for studies of genetic variation and its association with human disease, but epigenomic studies lack a similar reference. To address this need, the NIH Roadmap Epigenomics Consortium generated the largest collection so far of human epigenomes for primary cells and tissues. Here we describe the integrative analysis of 111 reference human epigenomes generated as part of the programme, profiled for histone modification patterns, DNA accessibility, DNA methylation and RNA expression. We establish global maps of regulatory elements, define regulatory modules of coordinated activity, and their likely activators and repressors. We show that disease- and trait-associated genetic variants are enriched in tissue-specific epigenomic marks, revealing biologically relevant cell types for diverse human traits, and providing a resource for interpreting the molecular basis of human disease. Our results demonstrate the central role of epigenomic information for understanding gene regulation, cellular differentiation and human disease.

Integrative analysis of 111 reference human epigenomes

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

The theory of phase-ordering dynamics that is the growth of order through domain coarsening when a system is quenched from the homogeneous phase into a broken-symmetry phase, is reviewed, with the emphasis on recent developments. Interest will focus on the scaling regime that develops at long times after the quench. How can one determine the growth laws that describe the time dependence of characteristic length scales, and what can be said about the form of the associated scaling functions? Particular attention will be paid to systems described by more complicated order parameters than the simple scalars usually considered, for example vector and tensor fields. The latter are needed, for example, to describe phase ordering in nematic liquid crystals, on which there have been a number of recent experiments. The study of topological defects (domain walls, vortices, strings and monopoles) provides a unifying framework for discussing coarsening in these different systems.

Theory of phase-ordering kinetics

Sphere Packings, Lattices and Groups

We present the simplest individual level model of predator-prey dynamics and show, via direct calculation, that it exhibits cycling behavior. The deterministic analogue of our model, recovered when the number of individuals is infinitely large, is the Volterra system (with density-dependent prey reproduction) which is well known to fail to predict cycles. This difference in behavior can be traced to a resonant amplification of demographic fluctuations which disappears only when the number of individuals is strictly infinite. Our results indicate that additional biological mechanisms, such as predator satiation, may not be necessary to explain observed predator-prey cycles in real (finite) populations.

/pdf/predator-prey-cycles-from-resonant-amplification-of-1opctpuol6.pdf

Predator-prey cycles from resonant amplification of demographic stochasticity.

The Gross-Pitaevskii approximation is a long-wavelength theory widely used to describe a variety of properties of dilute Bose condensates, in particular trapped alkali gases. We point out that for short-ranged repulsive interactions this theory fails in dimensions $d\ensuremath{\le}2$, and we propose the appropriate low-dimensional modifications, which have a universal form. For $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ we analyze density profiles in confining potentials, superfluid properties, solitons, and self-similar solutions.

https://arxiv.org/pdf/cond-mat/0002282.pdf

Low-dimensional bose liquids: beyond the gross-pitaevskii approximation

For many cancer types, incidence rises rapidly with age as an apparent power law, supporting the idea that cancer is caused by a gradual accumulation of genetic mutations. Similarly, the incidence of many infectious diseases strongly increases with age. Here, combining data from immunology and epidemiology, we show that many of these dramatic age-related increases in incidence can be modeled based on immune system decline, rather than mutation accumulation. In humans, the thymus atrophies from infancy, resulting in an exponential decline in T cell production with a half-life of ∼16 years, which we use as the basis for a minimal mathematical model of disease incidence. Our model outperforms the power law model with the same number of fitting parameters in describing cancer incidence data across a wide spectrum of different cancers, and provides excellent fits to infectious disease data. This framework provides mechanistic insight into cancer emergence, suggesting that age-related decline in T cell output is a major risk factor.

/pdf/thymic-involution-and-rising-disease-incidence-with-age-3rlojxvmsl.pdf

Thymic involution and rising disease incidence with age

We introduce a class of stochastic population models based on “patch dynamics.” The size of the patch may be varied, and this allows one to quantify the departures of these stochastic models from various mean-field theories, which are generally valid as the patch size becomes very large. These models may be used to formulate a broad range of biological processes in both spatial and nonspatial contexts. Here, we concentrate on two-species competition. We present both a mathematical analysis of the patch model, in which we derive the precise form of the competition mean-field equations (and their first-order corrections in the nonspatial case), and simulation results. These mean-field equations differ, in some important ways, from those which are normally written down on phenomenological grounds. Our general conclusion is that mean-field theory is more robust for spatial models than for a single isolated patch. This is due to the dilution of stochastic effects in a spatial setting resulting from repeated rescue events mediated by interpatch diffusion. However, discrete effects due to modest patch sizes lead to striking deviations from mean-field theory even in a spatial setting.

https://www.theory.physics.manchester.ac.uk/~ajm/mn04.pdf

Stochastic models in population biology and their deterministic analogs.

Recently, the Subcellular Element Model (SEM) has been introduced, primarily to compute the dynamics of large numbers of three-dimensional deformable cells in multicellular systems. Within this model framework, each cell is represented by a collection of elastically coupled elements, interacting with one another via short-range potentials, and dynamically updated using over-damped Langevin dynamics. The SEM can also be used to represent a single cell in more detail, by using a larger number of subcellular elements exclusively identified with that cell. We have tested whether, in this context, the SEM yields viscoelastic properties consistent with those measured on single living cells. Employing virtual methods of bulk rheology and microrheology we find that the SEM successfully captures many cellular rheological properties at intermediate time scales and moderate strains, including weak power law rheology. In its simplest guise, the SEM cannot describe long-time/large-strain cell responses. Capturing these cellular properties requires extensions of the SEM which incorporate active cytoskeletal rearrangement. Such extensions will be the subject of a future publication.

/pdf/modeling-cell-rheology-with-the-subcellular-element-model-5c4uln4c4i.pdf

Timothy J. Newman

Papers

Predator-prey cycles from resonant amplification of demographic stochasticity.

Low-dimensional bose liquids: beyond the gross-pitaevskii approximation

Thymic involution and rising disease incidence with age

Stochastic models in population biology and their deterministic analogs.

Modeling cell rheology with the Subcellular Element Model