Revision as of 17:46, 12 January 2017 by Mirzaev
AboutProf. David Bortz on microbial flocculation dynamics. I am also a huge soccer fan. I support Barça from La Liga and Manchester City from Barclays League.
Effect of flow and restructuring on microbial aggregate growth
This is a joint work with Eric P. Kightley. Our main goal in this project is to develop an agent-based model of microbial growth which also incorporates properties of the flow field. We plan to extend our current microscale model for more general flow fields. Our future research is guided by the following questions:
Answers to any of these questions would have broad-reaching impacts in many diverse areas such as water treatment, biofuel production, beer fermentation, etc. From the mathematical point of view, there would also be a substantial contribution on the microscale modeling of the microbial flocculation.
Identifying Conditional Probability Measures
Keywords: identification of probability measures, inverse problem, measure-dependent dynamical system, size-structured populations, flocculation, fragmentation, bacterial aggregates
In this work, we present and investigate an inverse problem for estimating the conditional probability measures from size-distribution measurements. We use the Prohorov metric (convergence in which is equivalent to weak convergence of measures) in a functional-analytic setting and show well-posedness of the inverse problem. We develop an approximation approach for computational implementation and show well-posedness of this approximate inverse problem. We also show the convergence of solutions to the approximate inverse problem to solutions of the original inverse problem. The primary contribution of this work is to extend this theory to conditional probability measures. We also illustrate that the flocculation dynamics of bacterial aggregates in suspension is one realization of systems satisfying the hypotheses in our framework.
Microbial Flocculation Dynamics
Keywords: Flocculation model, nonlinear evolution equations, principle of linearized stability, spectral analysis, structured populations dynamics, semigroup theory
Flocculation is the process whereby particles (i.e., flocs) in suspension reversibly combine and separate. The process is widespread in soft matter and aerosol physics as well as environmental science and engineering. We consider a general size-structured flocculation model, which describes the evolution of flocs in an aqueous environment. Our work provides a unified treatment for many size-structured models in the environmental, industrial, medical, and marine engineering literature. In particular, our model accounts for basic biological phenomena in a population of microorganisms including growth, death, sedimentation, predation, renewal, fragmentation and aggregation. Our central goal in this project is to rigorously investigate the long-term behavior of this generalized flocculation model. Using results from fixed point theory we derive conditions for the existence of continuous, non-trivial stationary solutions. We further apply the principle of linearized stability and semigroup compactness arguments to provide sufficient conditions for local exponential stability of stationary solutions as well as sufficient conditions for instability. The end results of this analytical development are relatively simple inequality-criteria which thus allows for the rapid evaluation of the existence and stability of a non-trivial stationary solution. To our knowledge, this work is the first to derive precise existence and stability criteria for such a generalized model. Lastly, we also provide an illustrating application of this criteria to several flocculation models. Here is an animation showing time evolution of microbial flocculation model:
Keywords: Laplacian dynamics, biochemical networks, synthesis and degradation, Matrix-Tree Theorem, insulin secretion
Analyzing qualitative behaviors of biochemical reactions using its associated network structure has proven useful in diverse branches of biology. As an extension of our previous work with Prof. Gunawardena, we introduce a graph-based framework to calculate steady state solutions of biochemical reaction networks with synthesis and degradation. Our approach is based on a labeled directed graph G and the associated system of linear non-homogeneous differential equations with first order degradation and zeroth order synthesis.
I have TA-ed the following courses so far:
"A problem well put is half solved" -- John Dewey