We present a new algorithm for the identification of bound regions from ChIP-Seq experiments. ChIP-Seq is a relatively new assay for measuring the interactions of proteins with DNA. The binding sites for a given protein in a genome are "peaks" in the data, which is given by an integer-valued height function defined on the genome. Our method for identifying statistically significant peaks is inspired by the notion of persistence in topological data analysis and provides a non-parametric approach that is robust to noise in experiments. Specifically, our method reduces the peak calling problem to the study of tree-based statistics derived from the data. The software T-PIC (Tree shape Peak Identification for ChIP-Seq) is available at http://math.berkeley.edu/~vhower/tpic.html and provides a fast and accurate solution for ChIP-Seq peak finding.

Chronic wound healing is a staggering public health problem, affecting 6.5 million individuals annually in the U.S. Ischemia, caused primarily by peripheral artery diseases, represents a major complicating factor in the healing process. In this talk, I will present a mathematical model of chronic wounds that represents the wounded tissue as a quasi-stationary Maxwell material, and incorporates the major biological processes involved in the wound closure. The model was formulated in terms of a system of partial differential equations with the surface of the open wound as a free boundary. Simulations of the model demonstrate how oxygen deficiency caused by ischemia limit macrophage recruitment to the wound-site and impair wound closure. The results are in tight agreement with recent experimental findings in a porcine model. I will also show analytical results of the model on the large-time asymptotic behavior of the free boundary under different ischemic conditions of the wound.

Correlations among neural spike times are found widely in the brain; they can be used to modulate or limit information in population coding, and open the possibility for cooperative coding of sensory inputs across neural populations. Correlations also introduce a daunting complexity; when every neuron is potentially correlated with every other, the amount of information needed to represent spiking activity grows exponentially with the number of cells.

In this talk I discuss recent work towards understanding how the structure and transfer of correlated activity is affected by both intrinsic neuron dynamics and network architecture. I first present an interesting and non-intuitive result about how the phase space structure of neural models --- specifically the bifurcation that mediates their transition from rest to firing --- affects their ability to transmit common signals. Second, I analyze the ability of pairwise maximum entropy models --- a powerful technique, borrowed from statistical mechanics, for representing spiking activity in a simpler way --- to perform on a broad class of feedforward circuits. This study provides an explanation for the surprising finding that responses in primate retinal ganglion cells are well-described by this model, even in cases where the circuit architecture seems likely to create a richer set of outputs (Shlens et al., J Neurosci, 2006; 2009; Schneidman et al., Nature, 2006), and takes some first steps towards understanding how the complexity of correlation structure depends on specific circuit mechanisms.

In this talk I will discuss a branching process model developed to study intra-tumor diversity (i.e. the variation amongst the cells within a single tumor). In this model changes in cellular fitness due to mutations are modeled by a bounded continuous random variable, so that the branching process has a continuous type-space. In the asymptotic (t-> infinity) regime, we study the growth rates of the population as well as several ecological measures of diversity in the tumor. In the latter part of the talk I will discuss an application of this model to study the evolution of diverse drug-resistant populations in Chronic Myeloid Leukemia.

We improved a computational model of leukemia development from stem cells to terminally differentiated cells by replacing the probabilistic, agent-based model of Roeder et al. (2006) with a system of partial differential equations (PDEs). The model is based on the relatively recent theory that cancer originates from cancer stem cells that reside in a microenvironment, called the stem cell niche. Depending on a stem cell's location within the stem cell niche, the stem cell may remain quiescent or begin proliferating. This theory states that leukemia (and potentially other cancers) is caused by the misregulation of the cycle of proliferation and quiescence within the stem cell niche.

Unlike the original agent-based model, which required seven hours per simulation, the PDE model could be numerically evaluated in a few minutes, and our numerical simulations showed that the PDE model closely replicated the average behavior of the original agent-based model. Furthermore, the PDE model was amenable to mathematical analysis, which revealed three modes of behavior: stability at 0 (cancer dies out), stability at a nonzero equilibrium (a scenario akin to chronic myelogenous leukemia), and periodic oscillations (a scenario akin to accelerated myelogenous leukemia).

The PDE formulation not only makes the model suitable for analysis, but also provides an effective mathematical framework for extending the model to include other aspects, such as the spatial distribution of stem cells within the niche.