# DE Seminars: Fall 2013

#### Monday September 16

Title |
Modeling modal transitions in diffusing populations |

Speaker |
Thomas I. Seidman Department of Mathematics and Statistics UMBC |

**Abstract:**

We consider a diffusing population as comprised of particles undergoing Brownian motion, each with its own history. For each such particle we then consider possible “state transitions” determined by crossing thresholds (hysteretic relay as in hybrid systems). Our objective here is to construct a continuum model of the resulting process as a reaction/diffusion system and then to show existence of “solutions” of this system. Technical difficulties arise here in resolving the concerns of hybrid systems (anomalous points and the possibility of Zeno phenomena) in a setting where one is tracking the collective effects on individual diffusing particles without being able to track their individual trajectories. For visualization, we think of an example of diffusing bacteria and nutrient in which each bacterium is reacting to its own experience of local nutrient concentration in switching between dormant and active modes.

#### Monday October 14

Title |
Computational Studies of Cardiac Excitation-Contraction Coupling: From Molecule to Arrhythmia |

Speaker |
Saleet Jafri George Mason University |

**Abstract:**

Calcium dynamics in the cardiac myocyte links the electrical excitation of the heart to contraction in a process known as excitation-contraction coupling. Dysfunction of critical calcium signaling proteins in heart is associated with lethal inherited cardiac arrhythmias. However, how the altered proteins lead to arrhythmias remains both unknown and controversial. We have used computational models to investigate fundamental mechanisms that underlie calcium-dependent arrhythmias, the same class of arrhythmias that follow myocardial infarction, heart failure and diverse genetic arrhythmic diseases. Even very common arrhythmias (one episode of sudden cardiac death in a month) are rare when normalized to the events occurring within a single cell over the period of a typical long experiment (e.g. one hour). Stochastic modeling, however, with the powerful computer clusters available and with our recent advances in computational algorithms, enable us to examine stochastic model systems over prolonged periods without missing the rare events. We start with the most elementary event of cardiac calcium release, the calcium spark, and construct stochastic models that explain mechanisms of calcium release termination, calcium homeostasis and the sarcoplasmic reticulum calcium leak, and the generation of arrhythmias from defects in calcium signaling. These insights begin to provide insight in to the normal and abnormal physiology of cardiac excitation-contraction coupling.

#### Monday October 28

Title |
Multiscale approximations in stochastic biochemical networks |

Speaker |
Hye-Won Kang Department of Mathematics and Statistics UMBC |

**Abstract:**

Stochastic effects may play an important role in mathematical modeling of biological and chemical processes in case the copy number of some component involved in the system is small. In this talk, stochastic modeling of biochemical networks with several examples is introduced and multiscale approximations of stochastic biochemical networks are suggested. Evolution of the network is modeled in terms of a continuous-time Markov jump process. Chemical reaction networks are generally large in size and they involve various scales in species numbers and reaction rate constants. The multiscale approximation method is introduced to reduce the network complexity and to derive limiting models with simple structure. Then, asymptotic behavior of the error between the full model and the limiting model is approximated. This is a joint work with Thomas G. Kurtz and Lea Popovic.

#### Monday November 4

Title |
Finite element method for linear elliptic problem in non-divergence form |

Speaker |
Wujun Zhang University of Maryland at College Park |

**Abstract:**

We design a finite element method for linear elliptic problem in non-divergence form, which satisfies the discrete maximum principle. The method ensures convergence to the viscosity solution. We develop a novel approach to carry out error estimation by discrete maximum principle. Applying this novel approach, namely, a discrete version of Alexandroff Bakelman Pucci maximum principle, we establish a rate of convergence of the discrete solution in the maximum norm.

#### Monday November 11

Title |
Moment Growth Bounds on Stochastic Population Processes |

Speaker |
Muruhan Rathinam Department of Mathematics and Statistics UMBC |

**Abstract:**

We consider the class of continuous time, time homogeneous Markov processes with the $N$ dimensional non-negative integer lattice as state space that have finitely many state independent jumpsize vectors. Such processes in general can be regarded as population processes modeling the vector copy number of $N$ different species undergoing $M$ different types of reaction/ interaction events. Typical examples of such processes are stochastically modeled chemical reactions, predator-prey models as well as epidemiological models. These processes are uniquely characterized by their “propensity” functions as well as their “stoichiometric vectors.”

Such processes often possess the property that given a deterministic initial condition the process always remains in a bounded region of the state space. We provide a necessary and sufficient condition on the stoichiometric matrix for this to hold. When the process is not bounded in the state space a natural question is whether finite moments of all orders exist. We provide two different sufficient conditions and one necessary condition for the existence of moments of all orders for all time $t>0$.

#### Monday November 18

Title |
Analysis of SI Models with Multiple Interacting Populations using Subpopulations with Forcing Terms |

Speaker |
Evelyn Thomas Department of Mathematics and Statistics UMBC |

**Abstract:**

As a system of differential equations describing an epidemiological system becomes large with multiple connections between subpopulations, the expressions for reproductive numbers and endemic equilibria become algebraically complicated, which makes drawing conclusions based on biological parameters difficult. We present a new method which deconstructs the larger system into smaller subsystems, captures the bridges between the smaller systems as external forces, and bounds the reproductive numbers of the full system in terms of reproductive numbers of the smaller systems, which are algebraically tractable. This method also allows us to analyze the size of the endemic equilibria.

#### Monday December 2

Title |
Atherosclerotic plaque development: strategies for modeling the growth and degradation of the fibrous cap |

Speaker |
Jonathan Bell Department of Mathematics and Statistics UMBC |

**Abstract:**

Cardiovascular disease is a leading cause of death in the US and many developed countries. Atherosclerosis is a major contributor to this disease profile. Atherosclerosis is an inflammatory disease of major arteries due to fatty lesions forming in arterial walls, causing stenosis (contracting blood flow) and thrombosis (blood clots, blockage). Certain lesions, called vulnerable plaques, are responsible for most deaths from atherosclerosis. The growth and degradation of these plaques is very dynamic, involving complex biochemical, hemodynamic, and mechanical interactions. But the present experimental means for studying arterial plaque development is limited, calling for augmenting such studies by mathematical modeling, analysis, and simulation. In this talk I will give a background to the biology and outline a strategy for model development, starting with an ODE model of principle chemical and cellular processes, and progressing to more complicated PDE models that include more mechanisms. At this stage little is proved, so the talk must be viewed as a possible roadmap for approaching a variety of questions.