THE GOAL: To use mathematical tools to understand qualitative and quantitative behavior of biophysical systems.

My main tools are ordinary and partial differential equations. Reduction of biocomplexity through dimensional analysis and perturbation theory produces systems ripe for bifurcation, phase portrait, and numerical analysis. Simulation of complex systems coupled with physical and mathematical interpretation leads to understanding of mechanistic principles.

Calcium Wave Initiation and Evolution in a Neuron

The hippocampus functions to solidify short terms memories into long term memories. Part of the mechanism to accomplish this is hypothesized to involve the malleability of individual neurons and the connections between them. Short term and long term changes are made locally at synapses on both the pre- and post-synaptic sides and include upregulation of protein machinery. Calcium plays a key role in many of these changes and understanding how calcium is triggered and moves within the cell is critical to understanding overall neuronal adaptation. Several experiments (Nakamura et al., 1999; Nakamura et al., 2000; Nakamura et al., 2002; Hong et al., 2004; Ross, 2005) have explored the subtle mechanisms in which calcium is released from the store in conjunction with synaptic stimulus to create large (>1μM) attenuating calcium waves. Properties of the calcium waves as they depend on timing between and concentrations of various chemicals such as inositol 1,4,5-triphosphate (IP3) and calcium itself are still being elucidated. We construct a model consisting of reaction-diffusion equations coupled to subordinate ordinary differential equations (Peercy, in preparation). This model is based on simplified geometry (see Figure 1) to show the initiation site of a calcium wave and its dependence on IP3 stimulated at the synapses in the oblique dendrites and diffused to the apical trunk.
Figure 1. Oblique/Apical Dendrite Model. Synaptic stimulation occurs at the distal ends of the oblique dendrite activating metabotropic glutamate receptors resulting in IP3 release which diffuses toward the apical trunk. IP3R are predominately located on the ER in the apical dendrite. A somatic electrode records and may be used to stimulate back propagating action potentials (BPAPs).
The study continues to address questions of propagation into the soma as well as questions of the properties related with repeated triggering events including ER refilling.

Oxygen Sensing in Escherichia coli

(with Steve Cox, Ka-Yiu San, George Bennett, Sagit Shalel-Levanon, Jiangfeng)

Oxygen deprivation is fatal for many organisms including ourselves. E. coli, however, have mechanisms to adjust and survive, if not thrive, in drastically varying oxygen concentrations. The sensor/regulator proteins and their many downstream influences are schematically realized in Figure 2.

Figure 2. Influence of Oxygen on E.coli. Along the electron transport chain (ETC) electrons are "backed up" affecting the reduced quinone pool (green). Reduced quinones release the autophosphorylation of ArcB (blue) which translates a phosphate group (red) to ArcA (white). Oligomerized ArcA acts as a transcription factor for many tricarboxylic acid cycle enzymes as well as terminal ETC components. ArcA inhibits low affinty cytochrome oxidase bo3 and activates high affinity cytochrome oxidase bd (purple). Fnr (yellow) also senses oxygen either through direct or indirect means and acts to inhibit both cytochrome oxidases.

My work with theoreticians and experimentalists in bioengineering and biochemistry has led to the construction of a ordinary differential equation model of the sensor/regulator systems Arc and Fnr, their upregulation of cytochrome oxidase bo3 and bd genes (cyo and cyd, respectively), and the susbsequent feedback onto the sensor/regulator trigger. We find interesting features of the dynamics such as hysteresis and oscillatory behavior which can be readily explained within the model framework (Peercy et al., in revision).

New data from mRNA measurements for cyo and cyd with enhance this model. We aim to expand the model to incorporate oxygen's affect on the regulation of enzymes for the tricarboxylic acid (TCA) cycle. In the process of doing this we will better understand the relationship between the TCA enzyme numbers and their function as measured in C13 labeled steady state flows.

Cardiac Arrhythmias Induced from the Ischemia Border Zone

(with Jim Keener)

Coronary artery disease often leads to an acute blockage at some level in the brached arteriole tree that feeds the heart muscle. The level of this blockage can correlate with the spatial extent of ischemia (lack of oxygen and nutrients with a loss of waste product removal). The effects of ischemia are graded depending on the extent of collateral blood flow establishing a border zone of ischemia. The variation in properties including potential differences across the border zone creates a current of injury. Such a current under appropriate conditions may lead to spontaneous oscillations, which in turn may lead to arrhythmias (Keener, 2003).
Figure 3. Schematic of Acute Cardiac Ischemia. A blockage in the coronary arterioles cuts off blood supply to downstream tissue. The most centrally located receptical of the halted blood flow has the highest degree of ischemia. Surrounding tissue is perfused at varying levels depending on collateral blood flow.
We have constructed both a coupled cell (ODE) model system (Peercy and Keener, 2005) and strip of tissue (1-D PDE) system (Peercy and Keener, in preparation) to study this phenomenon. We have established theory which relates stimulus of a single cell to that of the coupled system allowing for reduced experimental preparation to study aspects of that system. We are extending that theory to the strip of tissue.
Figure 4. Experimental Study of Ischemia on a Strip of Cardiac Tissue.