Jeffrey J. Fox

From topology to dynamics in biochemical networks.

Abstract formulations of the regulation of gene expression as random Boolean switching networks have been studied extensively over the past three decades. These models have been developed to make statistical predictions of the types of dynamics observed in biological networks based on network topology and interaction bias, p. For values of mean connectivity chosen to correspond to real biological networks, these models predict disordered dynamics. However, chaotic dynamics seems to be absent from the functioning of a normal cell. While these models use a fixed number of inputs for each element in the network, recent experimental evidence suggests that several biological networks have distributions in connectivity. We therefore study randomly constructed Boolean networks with distributions in the number of inputs, K, to each element. We study three distributions: delta function, Poisson, and power law (scale free). We analytically show that the critical value of the interaction bias parameter, p, above which steady state behavior is observed, is independent of the distribution in the limit of the number of elements N--> infinity. We also study these networks numerically. Using three different measures (types of attractors, fraction of elements that are active, and length of period), we show that finite, scale-free networks are more ordered than either the Poisson or delta function networks below the critical point. Thus the topology of scale-free biochemical networks, characterized by a wide distribution in the number of inputs per element, may provide a source of order in living cells. (c) 2001 American Institute of Physics.

Dynamic mechanism for conduction block in heart tissue

Previous work has shown that dynamic heterogeneity and conduction block can occur in homogeneous heart fibres during prolonged pacing at rapid rates. Here we investigated the mechanism for conduction block following the delivery of one to four premature stimuli using a coupled maps computer model of a one-dimensional canine heart fibre. The coupled maps model allowed us to identify the roles that velocity (V) restitution, action potential duration (D) restitution and cardiac memory (M) played in the development of spatial heterogeneity and conduction block. We found that the likelihood of conduction block could be reduced by three methods. (1) By altering the V restitution function so that conduction slowed at very short rest intervals (I). (2) By altering the D restitution function to reduce the sensitivity of D to changes in I. (3) By increasing the contribution of cardiac memory (M). Although the results of this study need to be confirmed experimentally, they suggest several potential interventions that may reduce the probability of arrhythmia induction.

Dynamic Mechanism for Initiation of Ventricular Fibrillation In Vivo

Background— Dynamically induced heterogeneities of repolarization may lead to wave-front destabilizations and initiation of ventricular fibrillation (VF). In a computer modeling study, we demonstrated that specific sequences of premature stimuli maximized dynamically induced spatial dispersion of refractoriness and predisposed the heart to the development of conduction block. The purpose of this study was to determine whether the computer model results pertained to the initiation of VF in dogs in vivo. Methods and Results— Monophasic action potentials were recorded from right and left ventricular endocardium in anesthetized beagle dogs (n=11) in vivo. Restitution of action potential duration and conduction time and the effective refractory period after delivery of the basic stimulus (S1) and each of 3 premature stimuli (S2, S3, S4) were determined at baseline and during verapamil infusion. The effective refractory period data were used to determine the interstimulus intervals for a sequence of 4 premature stimuli (S2S3S4S5=CLVF) for which the computer model predicted maximal spatial dispersion of refractoriness. Delivery of CLVF was associated with discordant action potential duration alternans and induction of VF in all dogs. Verapamil decreased spatial dispersion of refractoriness by reducing action potential duration and conduction time restitution in a dose-dependent fashion, effects that were associated with reduced inducibility of VF with CLVF. Conclusion— Maximizing dynamically induced spatial dispersion of repolarization appears to be an effective method for inducing VF. Reducing spatial dispersion of refractoriness by modulating restitution parameters can have an antifibrillatory effect in vivo.

Synchronization in Relaxation Oscillator Networks with Conduction Delays

We study locally coupled networks of relaxation oscillators with excitatory connections and conduction delays and propose a mechanism for achieving zero phase-lag synchrony. Our mechanism is based on the observation that different rates of motion along different nullclines of the system can lead to synchrony in the presence of conduction delays. We analyze the system of two coupled oscillators and derive phase compression rates. This analysis indicates how to choose nullclines for individual relaxation oscillators in order to induce rapid synchrony. The numerical simulations demonstrate that our analytical results extend to locally coupled networks with conduction delays and that these networks can attain rapid synchrony with appropriately chosen nullclines and initial conditions. The robustness of the proposed mechanism is verified with respect to different nullclines, variations in parameter values, and initial conditions.

Cross-talk between signaling pathways can generate robust oscillations in calcium and cAMP

Background To control and manipulate cellular signaling, we need to understand cellular strategies for information transfer, integration, and decision-making. A key feature of signal transduction is the generation of only a few intracellular messengers by many extracellular stimuli. Methodology/Principal Findings Here we model molecular cross-talk between two classic second messengers, cyclic AMP (cAMP) and calcium, and show that the dynamical complexity of the response of both messengers increases substantially through their interaction. In our model of a non-excitable cell, both cAMP and calcium concentrations can oscillate. If mutually inhibitory, cross-talk between the two second messengers can increase the range of agonist concentrations for which oscillations occur. If mutually activating, cross-talk decreases the oscillation range, but can generate ‘bursting’ oscillations of calcium and may enable better filtering of noise. Conclusion We postulate that this increased dynamical complexity allows the cell to encode more information, particularly if both second messengers encode signals. In their native environments, it is unlikely that cells are exposed to one stimulus at a time, and cross-talk may help generate sufficiently complex responses to allow the cell to discriminate between different combinations and concentrations of extracellular agonists.

Sharp Interface and Voltage Conservation in the Phase Field Method: Application to Cardiac Electrophysiology

We present a finite difference method for modeling the propagation of electrical waves in cardiac tissue using the cable equation with homogeneous Neumann boundary conditions. This method is novel in that it is derived by first discretizing the phase field method as described by Fenton et al. [Chaos, 15 (2005), p. 013502.] then taking a limit to recover a sharp interface. Our method provides for exact voltage conservation (up to floating point precision in application), can be used on arbitrary geometries, and can be implemented using only nearest neighbor coupling between grid points.

Computationally Efficient Strategy for Modeling the Effect of Ion Current Modifiers

Electrophysiological studies often seek to relate changes in ion current properties caused by a chemical modifier to changes in cellular properties. Therefore, quantifying concentration-dependent effects of modifiers on ion currents is a topic of importance. In this paper, we sought a mathematical method for using ion current data to predict the effect of several theoretical ion current modifiers on cellular and tissue properties that is computationally efficient without compromising predictive power. We focused on the current as an example case due to its link to long QT syndrome and arrhythmias, but these methods should be generally applicable to other electrophysiological studies. We compared predictions using a Markov model with mass action binding of the modifiers to specific conformational states of the channel to predictions generated by two simplified models. We investigated scaling conductance, and found that although this method produced predictions that agreed qualitatively with the more complicated model, it did not generate quantitatively consistent predictions for all modifiers tested. Our simulations showed that a more computationally efficient Hodgkin-Huxley model that incorporates the effect of modifiers through functional changes in the current produced quantitatively consistent predictions of concentration-dependent changes in cell and tissue properties for all modifiers tested.