Georgia Institute of Technology College of Sciences

Tissue remodeling involves the activation of proteases, enzymes capable of degrading the structural proteins of tissue and organs. The implications of the activation of these enzymes span all organ systems and therefore, many different disease pathologies, including cancer metastasis. This occurs when local proteolysis of the structural extracellular matrix allows for malignant cells to break free from the primary tumor and spread to other tissues. Mathematical models add value to this experimental system by explaining phenomena difficult to test at the wet lab bench and to make sense of complex interactions among the proteases or the intracellular signaling changes leading to their expression. The papain family of cysteine proteases, the cathepsins, is an understudied class of powerful collagenases and elastases implicated in extracellular matrix degradation that are secreted by macrophages and cancer cells and shown to be active in the slightly acidic tumor microenvironment. Due to the tight regulatory mechanisms of cathepsin activity and their instability outside of those defined spaces, detection of the active enzyme is difficult to precisely quantify, and therefore challenging to target therapeutically. Using valid assumptions that consider these complex interactions we are developing and validating a system of ordinary differential equations to calculate the concentrations of mature, active cathepsins in biological spaces. The system of reactions considers four enzymes (cathepsins B, K, L, and S, the most studied cathepsins with reaction rates available), three substrates (collagen IV, collagen I, and elastin) and one inhibitor (cystatin C) and comprise more than 30 differential equations with over 50 specified rate constants. Along with the mathematical model development, we have been developing new ways to quantify proteolytic activity to provide further inputs. This predictive model will be a useful tool in identifying the time scale and culprits of proteolytic breakdown leading to cancer metastasis and angiogenesis in malignant tumors.

One of the biggest hurdles in high performance computing today is the analysis of massive quantities of data. As the size of the datasets grows to petascale (and beyond), new techniques are needed to efficiently compute meaningful information from the raw data. Graph-based data (which is ubiquitous in social networks, biological interaction networks, etc) poses additional challenges due to the difficulty of parallelizing many common graph algorithms. A key component in success is the generation of "realistic" random data sets for testing and benchmarking new algorithms. The R-MAT graph generator introduced by Chakrabarti, Faloutsos, and Zhan (2004) offers a simple, fast method for generating very large directed graphs. One commonly held belief regarding graphs produced by R-MAT is that they are "scale free"; in other words, their degree distribution follows a power law as is observed in many real world networks. These properties have made R-MAT a popular choice for generating graphs for use in a variety of research disciplines including graph theoretic benchmarks, social network analysis, computational biology, and network monitoring. However, despite its wide usage and elegant, parsimonius design, our recent work provides the first rigorous mathematical analysis of the degree distributions of the generated graphs. Applying results from occupancy problems in probability theory, we derive exact expressions for the degree distributions and other parameters. We also prove that in the limit (as the number of vertices tends to infinity), graphs generated with R-MAT have degree distributions that can be expressed as a mixture of normal distributions. This talk will focus on the techniques used in solving this applied problem in terms of classical "ball and urn" results, including a minor extension of Chistyakov's theorem.

In 2006, my coadvisor Xiaoming Huo and his colleague published an annal of statistics paper which designs an asymptotically powerful testing algorithm to detect the potential curvilinear structure in a messy point cloud image. However, such an algorithm involves a membership threshold and a decision threshold which are not well defined in that paper because the distribution of LSP was unknown. Later on, Xiaoming's student Chen, Jihong found some connections between the distribution of LSP and the so-called Erdos-Renyi law. In some sense, the distribution of LSP is just a generalization of the Erdos-Renyi law. However this JASA paper of Chen, Jihong had some restrictions and only partially found out the distribution of LSP. In this talk, I will show the result of the JASA paper is actually very close to the distribution of LSP. However, these is still much potential work to do in order to strengthen this algorithm.

Finding ground states of spin glasses, a model of disordered materials, has a deep connection to many hard combinatorial optimization problems, such as satisfiability, maxcut, graph-bipartitioning, and coloring. Much insight has been gained for the combinatorial problems from the intuitive approaches developed in physics (such as replica theory and the cavity method), some of which have been proven rigorously recently. I present a treasure trove of numerical data obtained with heuristic methods that suggest a number conjectures, such as an equivalence between maxcut and bipartitioning for r-regular graphs, a simple relation for their optimal configurations as a function of degree r, and anomalous extreme-value fluctuations in a variety of models, hotly debated in physics currently. For some, such as those related to finite-size effects, not even a physics theory exists, for others theory exists that calls for rigorous methods.