MFBM Subgroup Contributed Talks

Eli Newby

The Pennsylvania State University

" Identifying Driver Nodes in Biological Networks using Subsets of the Feedback Vertex Set"

In network control theory, driving all the nodes in the Feedback Vertex Set (FVS) forces the network into one of its attractors (long-term dynamic behaviors), but the FVS is often composed of more nodes than can be realistically manipulated in a system (e.g., 1-3 nodes for intracellular networks). Thus, we developed a method to identify subsets of the FVS on Boolean models of intracellular networks using topological, dynamics-independent measures. We identified seven topological measures sorted into three categories — centrality measures, propagation measures, and cycle-based measures. Each measure was ranked and then evaluated against two dynamics-based metrics that measure ability of interventions to drive the system towards or away from their attractors: To Control and Away Control. After examining various biological networks, we found that the FVS subsets that ranked highest according to the propagation metrics could most effectively control the network. This result was independently corroborated on an array of different Boolean models of biological networks. Consequently, overriding the entire FVS is not required to drive a biological network to one of its attractors, and this method provides a way to reliably identify these FVS subsets without knowledge of the network's dynamics.

Xiaoyu Duan

University of Pittsburgh

"Parameter identification of linear-in-parameters systems from a single trajectory"

A fundamental problem in mathematical modeling is the determination of model parameter values from experimental data. In disease and immunological studies, repeated collection of data from a single subject is impossible, because the disease or the manipulations performed in the experiments are fatal to the subject, or permanently alter the subject's immune system. Therefore, parameter identification or estimation of nonlinear ODE systems from a single trajectory has been studied in my research. Many computational techniques for parameter estimation has been employed. However, it is also important to have rigorous, theoretical results, addressing the questions whether there exists solution, a unique solution, or no solution to the parameter estimation problem. If we view the models as forward mappings from parameter values to states of model variables, then the parameter identification/estimation from data can be formulated as the problem of inverting this mapping. Therefore this is an inverse problem and its solution is the set of parameters values. In particular, in my research, I focus on a certain type of nonlinear systems, which is linear-in-parameters system. I will use the two-dimensional Lotka-Volterra system as an example to show our results and explain difficulties in this research area.

Katherine Pearce

Department of Mathematics, North Carolina State University

"Modeling and parameter subset selection for fibrin polymerization kinetics with applications to wound healing"

During hemostasis in wound healing, vascular injury leads to endothelial cell damage, initiation of a coagulation cascade involving platelets, and formation of a fibrin-rich clot. Activation of the protease thrombin occurs and soluble fibrinogen is converted into an insoluble polymerized fibrin network. Fibrin polymerization is critical for bleeding cessation and subsequent stages of wound healing. We present a cooperative enzyme kinetics model for in vitro fibrin matrix polymerization capturing dynamic interactions among fibrinogen, thrombin, fibrin and intermediate complexes. A tailored parameter subset selection technique is implemented to evaluate parameter identifiability for a representative dataset for fibrin accumulation. Our approach is based on systematic analysis of the eigenvalues and eigenvectors of the information matrix for the quantity of interest fibrin matrix via optimization, based on a least squares objective function. Capabilities of this approach to decrease the objective cost and integrate non-overlapping subsets of the data to enhance the evaluation of parameter identifiability and aid in model reduction are also demonstrated. These findings illustrate the high degree of information within a single fibrin accumulation curve using a tailored model and parameter subset selection approach that can improve optimization and reduce model complexity.

Yue Liu

University of Oxford

"Organisation of diffusion-driven stripe formation in expanding domains"

In certain biological systems, such as the plumage pattern of birds and stripes on certain species of fishes, pattern formation take place behind a wave of competency. For these systems, one needs to consider the patterns that form when a particular type of growth -- apical growth -- is included. In this study, we use a particular type of partial differential equation model, known as a Turing diffusion-driven instability model, to study pattern formation on apically growing domains, under a variety of rates of growth. Numerical simulations show that in one spatial dimension a slower growth rate drives a sequence of peak splittings in the pattern, whereas a higher growth rate leads to peak insertions. In two spatial dimensions, we observe stripes that are either perpendicular or parallel to the moving boundary under slow or fast growth rates, respectively. To understand this phenomenon, we use stability and bifurcation analysis to understand the process of selection of stripes or spots. Finally, we argue that there is a correspondence between the one- and two-dimensional phenomena, and that apical growth can account for the pattern organization observed in many biological systems.