Georgia Institute of Technology College of Sciences

Almost all models for complex fluids can be fitted into the energetic variational framework. The advantage of the approach is the revealing/focus of the competition between the kinetic energy and the internal "elastic" energies. In this talk, I will discuss two very different engineering problems: free interface motion in Newtonian fluids and viscoelastic materials. We will illustrate the underlying connections between the problems and their distinct properties. Moreover, I will present the analytical results concerning the existence of near equilibrium solutions of these problems.

Nonnegative Matrix Factorization (NMF) has attracted much attention during the past decade as a dimension reduction method in machine learning and data analysis. NMF provides a lower rank approximation of a nonnegative high dimensional matrix by factors whose elements are also nonnegative. Numerous success stories were reported in application areas including text clustering, computer vision, and cancer class discovery. In this talk, we present novel algorithms for NMF and NTF (nonnegative tensor factorization) based on the alternating non-negativity constrained least squares (ANLS) framework. Our new algorithm for NMF is built upon the block principal pivoting method for the non-negativity constrained least squares problem that overcomes some limitations of the classical active set method. The proposed NMF algorithm can naturally be extended to obtain highly efficient NTF algorithm for PARAFAC (PARAllel FACtor) model. Our algorithms inherit the convergence theory of the ANLS framework and can easily be extended to other NMF formulations such as sparse NMF and NTF with L1 norm constraints. Comparisons of algorithms using various data sets show that the proposed new algorithms outperform existing ones in computational speed as well as the solution quality. This is a joint work with Jingu Kim and Krishnakumar Balabusramanian.

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.

How do animals move without legs? In this experimental and theoretical study, we investigate the slithering of snakes on flat surfaces. Previous studies of slithering have rested on the assumption that snakes slither by pushing laterally against rocks and branches. In this combined experimental and theoretical study, we develop a model for slithering locomotion by observing snake motion kinematics and experimentally measuring the friction coefficients of snake skin. Our predictions of body speed show good agreement with observations, demonstrating that snake propulsion on flat ground, and possibly in general, relies critically on the frictional anisotropy of their scales. We also highlight the importance of the snake's dynamically redistributing its weight during locomotion in order to improve speed and efficiency. We conclude with an overview of our experimental observations of other methods of propulsion by snakes, including sidewinding and a unidirectional accordion-like mode.