Insights into pathological mechanisms and interventions revealed by analyzing a mathematical model for cone metabolism.

This work analyzes a mathematical model for the metabolic dynamics of a cone photoreceptor, which is the first model to account for energy generation from fatty acids oxidation of shed photoreceptor outer segments (POS). Multiple parameter bifurcation analysis shows that joint variations in external glucose, the efficiency of glucose transporter 1 (GLUT1), lipid utilization for POS renewal, and oxidation of fatty acids affect the cone's metabolic vitality and its capability to adapt under glucose-deficient conditions. The analysis further reveals that when glucose is scarce, cone viability cannot be sustained by only fueling energy production in the mitochondria, but it also requires supporting anabolic processes to create lipids necessary for cell maintenance and repair. In silico experiments are used to investigate how the duration of glucose deprivation impacts the cell without and with a potential GLUT1 or oxidation of fatty acids intervention as well as a dual intervention. The results show that for prolonged duration of glucose deprivation, the cone metabolic system does not recover with higher oxidation of fatty acids and requires greater effectiveness of GLUT1 to recover. Finally, time-varying global sensitivity analysis (GSA) is applied to assess the sensitivity of the model outputs of interest to changes and uncertainty in the parameters at specific times. The results reveal a critical temporal window where there would be more flexibility for interventions to rescue a cone cell from the detrimental consequences of glucose shortage.

Thermal Performance Curves of Multiple Isolates of Batrachochytrium dendrobatidis, a Lethal Pathogen of Amphibians.

Emerging infectious disease is a key factor in the loss of amphibian diversity. In particular, the disease chytridiomycosis has caused severe declines around the world. The lethal fungal pathogen that causes chytridiomycosis, Batrachochytrium dendrobatidis (Bd), has affected amphibians in many different environments. One primary question for researchers grappling with disease-induced losses of amphibian biodiversity is what abiotic factors drive Bd pathogenicity in different environments. To study environmental influences on Bd pathogenicity, we quantified responses of Bd phenotypic traits (e.g., viability, zoospore densities, growth rates, and carrying capacities) over a range of environmental temperatures to generate thermal performance curves. We selected multiple Bd isolates that belong to a single genetic lineage but that were collected across a latitudinal gradient. For the population viability, we found that the isolates had similar thermal optima at 21°C, but there was considerable variation among the isolates in maximum viability at that temperature. Additionally, we found the densities of infectious zoospores varied among isolates across all temperatures. Our results suggest that temperatures across geographic point of origin (latitude) may explain some of the variation in Bd viability through vertical shifts in maximal performance. However, the same pattern was not evident for other reproductive parameters (zoospore densities, growth rates, fecundity), underscoring the importance of measuring multiple traits to understand variation in pathogen responses to environmental conditions. We suggest that variation among Bd genetic variants due to environmental factors may be an important determinant of disease dynamics for amphibians across a range of diverse environments.

Stochastic shielding and edge importance for Markov chains with timescale separation.

Nerve cells produce electrical impulses ("spikes") through the coordinated opening and closing of ion channels. Markov processes with voltage-dependent transition rates capture the stochasticity of spike generation at the cost of complex, time-consuming simulations. Schmandt and Galán introduced a novel method, based on the stochastic shielding approximation, as a fast, accurate method for generating approximate sample paths with excellent first and second moment agreement to exact stochastic simulations. We previously analyzed the mathematical basis for the method's remarkable accuracy, and showed that for models with a Gaussian noise approximation, the stationary variance of the occupancy at each vertex in the ion channel state graph could be written as a sum of distinct contributions from each edge in the graph. We extend this analysis to arbitrary discrete population models with first-order kinetics. The resulting decomposition allows us to rank the "importance" of each edge's contribution to the variance of the current under stationary conditions. In most cases, transitions between open (conducting) and closed (non-conducting) states make the greatest contributions to the variance, but there are exceptions. In a 5-state model of the nicotinic acetylcholine receptor, at low agonist concentration, a pair of "hidden" transitions (between two closed states) makes a greater contribution to the variance than any of the open-closed transitions. We exhaustively investigate this "edge importance reversal" phenomenon in simplified 3-state models, and obtain an exact formula for the contribution of each edge to the variance of the open state. Two conditions contribute to reversals: the opening rate should be faster than all other rates in the system, and the closed state leading to the opening rate should be sparsely occupied. When edge importance reversal occurs, current fluctuations are dominated by a slow noise component arising from the hidden transitions.

Behavior Responses to Chemical and Optogenetic Stimuli in Drosophila Larvae.

An animal's ability to navigate an olfactory environment is critically dependent on the activities of its first-order olfactory receptor neurons (ORNs). While considerable research has focused on ORN responses to odorants, the mechanisms by which olfactory information is encoded in the activities of ORNs and translated into navigational behavior remain poorly understood. We sought to determine the contributions of most Drosophila melanogaster larval ORNs to navigational behavior. Using odorants to activate ORNs and a larval tracking assay to measure the corresponding behavioral response, we observed that larval ORN activators cluster into four groups based on the behavior responses elicited from larvae. This is significant because it provides new insights into the functional relationship between ORN activity and behavioral response. Subsequent optogenetic analyses of a subset of ORNs revealed previously undescribed properties of larval ORNs. Furthermore, our results indicated that different temporal patterns of ORN activation elicit different behavioral outputs: some ORNs respond to stimulus increments while others respond to stimulus decrements. These results suggest that the ability of ORNs to encode temporal patterns of stimulation increases the coding capacity of the olfactory circuit. Moreover, the ability of ORNs to sense stimulus increments and decrements facilitates instantaneous evaluations of concentration changes in the environment. Together, these ORN properties enable larvae to efficiently navigate a complex olfactory environment. Ultimately, knowledge of how ORN activity patterns and their weighted contributions influence odor coding may eventually reveal how peripheral information is organized and transmitted to subsequent layers of a neural circuit.

Measuring edge importance: a quantitative analysis of the stochastic shielding approximation for random processes on graphs.

Mathematical models of cellular physiological mechanisms often involve random walks on graphs representing transitions within networks of functional states. Schmandt and Galán recently introduced a novel stochastic shielding approximation as a fast, accurate method for generating approximate sample paths from a finite state Markov process in which only a subset of states are observable. For example, in ion-channel models, such as the Hodgkin-Huxley or other conductance-based neural models, a nerve cell has a population of ion channels whose states comprise the nodes of a graph, only some of which allow a transmembrane current to pass. The stochastic shielding approximation consists of neglecting fluctuations in the dynamics associated with edges in the graph not directly affecting the observable states. We consider the problem of finding the optimal complexity reducing mapping from a stochastic process on a graph to an approximate process on a smaller sample space, as determined by the choice of a particular linear measurement functional on the graph. The partitioning of ion-channel states into conducting versus nonconducting states provides a case in point. In addition to establishing that Schmandt and Galán's approximation is in fact optimal in a specific sense, we use recent results from random matrix theory to provide heuristic error estimates for the accuracy of the stochastic shielding approximation for an ensemble of random graphs. Moreover, we provide a novel quantitative measure of the contribution of individual transitions within the reaction graph to the accuracy of the approximate process.

Hormone secretion in transgenic rats and electrophysiological activity in their gonadotropin releasing-hormone neurons.

Expression of GFP in GnRH neurons has allowed for studies of individual GnRH neurons. We have demonstrated previously the preservation of physiological function in male GnRH-GFP mice. In the present study, we confirm using biocytin-filled GFP-positive neurons in the hypothalamic slice preparation that GFP-expressing somata, axons, and dendrites in hypothalamic slices from GnRH-GFP rats are GnRH1 peptide positive. Second, we used repetitive sampling to study hormone secretion from GnRH-GFP transgenic rats in the homozygous, heterozygous, and wild-type state and between transgenic and Wistar males after ~4 yr of backcrossing. Parameters of hormone secretion were not different between the three genetic groups or between transgenic males and Wistar controls. Finally, we performed long-term recording in as many GFP-identified GnRH neurons as possible in hypothalamic slices to determine their patterns of discharge. In some cases, we obtained GnRH neuronal recordings from individual males in which blood samples had been collected the previous day. Activity in individual GnRH neurons was expressed as total quiescence, a continuous pattern of firing of either low or relatively high frequencies or an intermittent pattern of firing. In males with both intensive blood sampling (at 6-min intervals) and recordings from their GnRH neurons, we analyzed the activity of GnRH neurons with intermittent activity above 2 Hz using cluster analysis on both data sets. The average number of pulses was 3.9 ± 0.6/h. The average number of episodes of firing was 4.0 ± 0.6/h. Therefore, the GnRH pulse generator may be maintained in the sagittal hypothalamic slice preparation.

Waiting for two mutations: with applications to regulatory sequence evolution and the limits of Darwinian evolution.

Results of Nowak and collaborators concerning the onset of cancer due to the inactivation of tumor suppressor genes give the distribution of the time until some individual in a population has experienced two prespecified mutations and the time until this mutant phenotype becomes fixed in the population. In this article we apply these results to obtain insights into regulatory sequence evolution in Drosophila and humans. In particular, we examine the waiting time for a pair of mutations, the first of which inactivates an existing transcription factor binding site and the second of which creates a new one. Consistent with recent experimental observations for Drosophila, we find that a few million years is sufficient, but for humans with a much smaller effective population size, this type of change would take > 100 million years. In addition, we use these results to expose flaws in some of Michael Behe's arguments concerning mathematical limits to Darwinian evolution.

Adaptive evolution drives the diversification of zinc-finger binding domains.

The human genome is estimated to contain 700 zinc-finger genes, which perform many key functions, including regulating transcription. The dramatic increase in the number of these genes as we move from yeast to C. elegans to Drosophila and to humans, as well as the clustered organization of these genes in humans, suggests that gene duplication has played an important role in expanding this family of genes. Using likelihood methods developed by Yang and parsimony methods introduced by Suzuki and Gojobori, we have investigated four clusters of zinc-finger genes on human chromosome 19 and found evidence that positive selection was involved in diversifying the family of zinc-finger binding motifs.