We study the voter model on heterogeneous graphs. We exploit the nonconservation of the magnetization to characterize how consensus is reached. For a network of N nodes with an arbitrary but uncorrelated degree distribution, the mean time to reach consensus T(N) scales as Nmu(2)1/mu(2), where mu(k) is the kth moment of the degree distribution. For a power-law degree distribution n(k) approximately k(-nu), T(N) thus scales as N for nu > 3, as N/ln(N for nu = 3, as N((2nu-4)/(nu-1)) for 2 < nu < 3, as (lnN)2 for nu = 2, and as omicron(1) for nu < 2. These results agree with simulation data for networks with both uncorrelated and correlated node degrees.

We study simple interacting particle systems on heterogeneous networks, including the voter model and the invasion process. These are both two-state models in which in an update event an individual changes state to agree with a neighbor. For the voter model, an individual "imports" its state from a randomly-chosen neighbor. Here the average time T N to reach consensus for a network of N nodes with an uncorrelated degree distribution scales as , where μ k is the k th moment of the degree distribution. Quick consensus thus arises on networks with broad degree distributions. We also identify the conservation law that characterizes the route by which consensus is reached. Parallel results are derived for the invasion process, in which the state of an agent is "exported" to a random neighbor. We further generalize to biased dynamics in which one state is favored. The probability for a single fitter mutant located at a node of degree k to overspread the population-the fixation probability-is proportional to k for the voter model and to 1/k for the invasion process.

The evolution of two species with different fitness is investigated on degree-heterogeneous graphs. The population evolves either by one individual dying and being replaced by the offspring of a random neighbor (voter model dynamics) or by an individual giving birth to an offspring that takes over a random neighbor node (invasion process dynamics). The fixation probability for one species to take over a population of N individuals depends crucially on the dynamics and on the local environment. Starting with a single fitter mutant at a node of degree k, the fixation probability is proportional to k for voter model dynamics and to 1/k for invasion process dynamics.

We study the mean time for a random walk to traverse between two arbitrary sites of the Erdős-Renyi random graph. We develop an effective medium approximation that predicts that the mean first-passage time between pairs of nodes, as well as all moments of this first-passage time, are insensitive to the fraction p of occupied links. This prediction qualitatively agrees with numerical simulations away from the percolation threshold. Near the percolation threshold, the statistically meaningful quantity is the mean transit rate, namely, the inverse of the first-passage time. This rate varies non-monotonically with p near the percolation transition. Much of this behavior can be understood by simple heuristic arguments.

Classic semiquantitative proteomic methods have shown that all organisms respond to a mild heat shock by an apparent massive accumulation of a small set of proteins, named heat-shock proteins (HSPs) and a concomitant slowing down in the synthesis of the other proteins. Yet unexplained, the increased levels of HSP messenger RNAs (mRNAs) may exceed 100 times the ensuing relative levels of HSP proteins. We used here high-throughput quantitative proteomics and targeted mRNA quantification to estimate in human cell cultures the mass and copy numbers of the most abundant proteins that become significantly accumulated, depleted, or unchanged during and following 4 h at 41 °C, which we define as mild heat shock. This treatment caused a minor across-the-board mass loss in many housekeeping proteins, which was matched by a mass gain in a few HSPs, predominantly cytosolic HSPCs (HSP90s) and HSPA8 (HSC70). As the mRNAs of the heat-depleted proteins were not significantly degraded and less ribosomes were recruited by excess new HSP mRNAs, the mild depletion of the many housekeeping proteins during heat shock was attributed to their slower replenishment. This differential protein expression pattern was reproduced by isothermal treatments with Hsp90 inhibitors. Unexpectedly, heat-treated cells accumulated 55 times more new molecules of HSPA8 (HSC70) than of the acknowledged heat-inducible isoform HSPA1A (HSP70), implying that when expressed as net copy number differences, rather than as mere “fold change” ratios, new biologically relevant information can be extracted from quantitative proteomic data. Raw data are available via ProteomeXchange with identifier PXD001666.Electronic supplementary materialThe online version of this article (doi:10.1007/s12192-015-0583-2) contains supplementary material, which is available to authorized users.