Bree Cummins

A Spatial Model of Mosquito Host-Seeking Behavior

Mosquito host-seeking behavior and heterogeneity in host distribution are important factors in predicting the transmission dynamics of mosquito-borne infections such as dengue fever, malaria, chikungunya, and West Nile virus. We develop and analyze a new mathematical model to describe the effect of spatial heterogeneity on the contact rate between mosquito vectors and hosts. The model includes odor plumes generated by spatially distributed hosts, wind velocity, and mosquito behavior based on both the prevailing wind and the odor plume. On a spatial scale of meters and a time scale of minutes, we compare the effectiveness of different plume-finding and plume-tracking strategies that mosquitoes could use to locate a host. The results show that two different models of chemotaxis are capable of producing comparable results given appropriate parameter choices and that host finding is optimized by a strategy of flying across the wind until the odor plume is intercepted. We also assess the impact of changing the level of host aggregation on mosquito host-finding success near the end of the host-seeking flight. When clusters of hosts are more tightly associated on smaller patches, the odor plume is narrower and the biting rate per host is decreased. For two host groups of unequal number but equal spatial density, the biting rate per host is lower in the group with more individuals, indicative of an attack abatement effect of host aggregation. We discuss how this approach could assist parameter choices in compartmental models that do not explicitly model the spatial arrangement of individuals and how the model could address larger spatial scales and other probability models for mosquito behavior, such as Lévy distributions.

Computation of three-dimensional Brinkman flows using regularized methods

The Brinkman equations of fluid motion are a model of flows in a porous medium. We develop the exact solution of the Brinkman equations for three-dimensional incompressible flow driven by regularized forces. Two different approaches to the regularization are discussed and compared on test problems. The regularized Brinkman model is also applied to the unsteady Stokes equation for oscillatory flows since the latter leads to the Brinkman equations with complex permeability parameter. We provide validation studies of the method based on the flow and drag of a solid sphere translating in a Brinkman medium and the flow inside a cylindrical channel of circular cross-section. We present a numerical example of a swimming organism in a Brinkman flow which shows that the maximum swimming speed is obtained with a small but non-zero value of the porosity. We also demonstrate that unsteady Stokes flows with oscillatory forcing fall within the same framework and are computed with the same method by applying it to the motion of the oscillating feeding appendage of a copepod.

Combinatorial Representation of Parameter Space for Switching Networks

We describe the theoretical and computational framework for the Dynamic Signatures for Genetic Regulatory Network ( DSGRN) database. The motivation stems from urgent need to understand the global dynamics of biologically relevant signal transduction/gene regulatory networks that have at least 5 to 10 nodes, involve multiple interactions, and decades of parameters. The input to the database computations is a regulatory network, i.e. a directed graph with edges indicating up or down regulation. A computational model based on switching networks is generated from the regulatory network. The phase space dimension of this model equals the number of nodes and the associated parameter space consists of one parameter for each node (a decay rate), and three parameters for each edge (low level of expression, high level of expression, and threshold at which expression levels change). Since the nonlinearities of switching systems are piece-wise constant, there is a natural decomposition of phase space into cells from which the dynamics can be described combinatorially in terms of a state transition graph. This in turn leads to a compact representation of the global dynamics called an annotated Morse graph that identifies recurrent and nonrecurrent dynamics. The focus of this paper is on the construction of a natural computable finite decomposition of parameter space into domains where the annotated Morse graph description of dynamics is constant. We use this decomposition to construct an SQL database that can be effectively searched for dynamical signatures such as bistability, stable or unstable oscillations, and stable equilibria. We include two simple 3-node networks to provide small explicit examples of the type of information stored in the DSGRN database. To demonstrate the computational capabilities of this system we consider a simple network associated with p53 that involves 5 nodes and a 29-dimensional parameter space.

Assessing the mechanical response of groups of arthropod filiform flow sensors

Many arthropod filiform flow sensors contain hundreds of sensilla that respond to complex natural airflow stimuli. The combination of complex morphology and richness of natural stimuli present a significant challenge for modeling these sensory apparatuses. We survey advances in our understanding over the last five years of the interactions between filiform hairs immersed in airflow. We then discuss a general characterization of filiform hair response based on our work modeling the cricket cercal system, which utilizes a newly developed unsteady Stokes model to investigate the response of a group of hairs to temporally complex signals.

A Refined Model of Viscous Coupling Between Filiform Hairs in the Cricket Cercal System

A model for the viscous interaction between filiform hairs on the cricket cercus was previously introduced by Cummins et al. [1]. This model simulates hair movement for a small group of arbitrarily positioned hairs stimulated by axial air flow along the cercus by calculating the perturbed boundary layer surrounding the hairs. In order to solve the perturbation calculation, Cummins et al. [1] introduce a simplification. However, this approximation introduces non-negligible error into the boundary conditions of the problem. A method of iterative refinement is presented in this paper that results in a more accurate approximation to the perturbed boundary layer. The changes to the predictions given in the previous paper are discussed.Copyright

Convergence Properties of Posttranslationally Modified Protein–Protein Switching Networks with Fast Decay Rates

A significant conceptual difficulty in the use of switching systems to model regulatory networks is the presence of so-called “black walls,” co-dimension 1 regions of phase space with a vector field pointing inward on both sides of the hyperplane. Black walls result from the existence of direct negative self-regulation in the system. One biologically inspired way of removing black walls is the introduction of intermediate variables that mediate the negative self-regulation. In this paper, we study such a perturbation. We replace a switching system with a higher-dimensional switching system with rapidly decaying intermediate proteins, and compare the dynamics between the two systems. We find that the while the individual solutions of the original system can be approximated for a finite time by solutions of a sufficiently close perturbed system, there are always solutions that are not well approximated for any fixed perturbation. We also study a particular example, where global basins of attraction of the perturbed system have a strikingly different form than those of the original system. We perform this analysis using techniques that are adapted to dealing with non-smooth systems.

Database of Dynamic Signatures Generated by Regulatory Networks (DSGRN)

We present a computational tool DSGRN for exploring network dynamics across the global parameter space for switching model representations of regulatory networks. This tool provides a finite partition of parameter space such that for each region in this partition a global description of the dynamical behavior of a network is given via a directed acyclic graph called a Morse graph. Using this method, parameter regimes or entire networks may be rejected as viable models for representing the underlying regulatory mechanisms.

Global dynamics for switching systems and their extensions by linear differential equations

Switching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.

The ecology of movement and behaviour: a saturated tripartite network for describing animal contacts

Ecologists regularly use animal contact networks to describe interactions underlying pathogen transmission, gene flow, and information transfer. However, empirical descriptions of contact often overlook some features of individual movement, and decisions about what kind of network to use in a particular setting are commonly ad hoc. Here, we relate individual movement trajectories to contact networks through a tripartite network model of individual, space, and time nodes. Most networks used in animal contact studies (e.g. individual association networks, home range overlap networks, and spatial networks) are simplifications of this tripartite model. The tripartite structure can incorporate a broad suite of alternative ecological metrics like home range sizes and patch occupancy patterns into inferences about contact network metrics such as modularity and degree distribution. We demonstrate the models utility with two simulation studies using alternative forms of ecological data to constrain the tripartite networks structure and inform expectations about the harder-to-measure metrics related to contact.

Identifying robust hysteresis in networks

We present a new modeling and computational tool that computes rigorous summaries of network dynamics over large sets of parameter values. These summaries, organized in a database, can be searched for observed dynamics, e.g., bistability and hysteresis, to discover parameter regimes over which they are supported. We illustrate our approach on several networks underlying the restriction point of the cell cycle in humans and yeast. We rank networks by how robustly they support hysteresis, which is the observed phenotype. We find that the best 6-node human network and the yeast network share similar topology and robustness of hysteresis, in spite of having no homology between the corresponding nodes of the network. Our approach provides a new tool linking network structure and dynamics.