Graph Theoretical Model of a Sensorimotor Connectome in Zebrafish

Mapping the detailed connectivity patterns (connectomes) of neural circuits is a central goal of neuroscience. The best quantitative approach to analyzing connectome data is still unclear but graph theory has been used with success. We present a graph theoretical model of the posterior lateral line sensorimotor pathway in zebrafish. The model includes 2,616 neurons and 167,114 synaptic connections. Model neurons represent known cell types in zebrafish larvae, and connections were set stochastically following rules based on biological literature. Thus, our model is a uniquely detailed computational representation of a vertebrate connectome. The connectome has low overall connection density, with 2.45% of all possible connections, a value within the physiological range. We used graph theoretical tools to compare the zebrafish connectome graph to small-world, random and structured random graphs of the same size. For each type of graph, 100 randomly generated instantiations were considered. Degree distribution (the number of connections per neuron) varied more in the zebrafish graph than in same size graphs with less biological detail. There was high local clustering and a short average path length between nodes, implying a small-world structure similar to other neural connectomes and complex networks. The graph was found not to be scale-free, in agreement with some other neural connectomes. An experimental lesion was performed that targeted three model brain neurons, including the Mauthner neuron, known to control fast escape turns. The lesion decreased the number of short paths between sensory and motor neurons analogous to the behavioral effects of the same lesion in zebrafish. This model is expandable and can be used to organize and interpret a growing database of information on the zebrafish connectome

Applications of Uncertainty Quantification to Coagulation Biology

Blood coagulation is a complex system comprised of numerous biochemical reactions. Due to this complexity, mathematical modeling has been used to increase the overall understanding of the system as a whole, determine previously unknown mechanisms, and to predict system responses. These models, however, may involve uncertainty in both parameter values and kinetic schemes that describe the reactions; this dissertation examines two such ideas. First, we examine the interactions between a specific coagulation factor, FXa, and an experimental tool designed to measure its action, a chromogenic substrate. Second, we examine a more complex mathematical model in regards to its parametric uncertainty. Chapter 1 gives a background on the mathematical tools used in this dissertation and necessary for uncertainty quantification (UQ) and an overview of the two aforementioned systems. In Chapter 2 we demonstrate how an application of UQ identifies a new model for product inhibition between FXa and its chromogenic substrate, which is validated experimentally. In Chapter 3 we conduct an extensive local and global sensitivity analysis for a mathematical model of flow-mediated blood coagulation. We determined that for many cases a local analysis is sufficient to understand the uncertainty in the model’s output, but that for certain cases there are classes of parameters that exhibit strong synergistic behavior, and so a global method that is capable of resolving interaction effects is necessary. These results motivated the work in Chapter 4 where we used global sensitivity analysis on a mathematical model to identify a novel mechanism for recovering a normal clotting response in hemophilia A; the potential mechanism was further supported by experimental validation. Chapter 5 summarizes the conclusions from the preceding chapters and presents ongoing work relating to the two projects

Applications of Uncertainty Quantification to Coagulation Biology

Blood coagulation is a complex system comprised of numerous biochemical reactions. Due to this complexity, mathematical modeling has been used to increase the overall understanding of the system as a whole, determine previously unknown mechanisms, and to predict system responses. These models, however, may involve uncertainty in both parameter values and kinetic schemes that describe the reactions; this dissertation examines two such ideas. First, we examine the interactions between a specific coagulation factor, FXa, and an experimental tool designed to measure its action, a chromogenic substrate. Second, we examine a more complex mathematical model in regards to its parametric uncertainty. Chapter 1 gives a background on the mathematical tools used in this dissertation and necessary for uncertainty quantification (UQ) and an overview of the two aforementioned systems. In Chapter 2 we demonstrate how an application of UQ identifies a new model for product inhibition between FXa and its chromogenic substrate, which is validated experimentally. In Chapter 3 we conduct an extensive local and global sensitivity analysis for a mathematical model of flow-mediated blood coagulation. We determined that for many cases a local analysis is sufficient to understand the uncertainty in the model’s output, but that for certain cases there are classes of parameters that exhibit strong synergistic behavior, and so a global method that is capable of resolving interaction effects is necessary. These results motivated the work in Chapter 4 where we used global sensitivity analysis on a mathematical model to identify a novel mechanism for recovering a normal clotting response in hemophilia A; the potential mechanism was further supported by experimental validation. Chapter 5 summarizes the conclusions from the preceding chapters and presents ongoing work relating to the two projects

Variation in thrombin generation as a result of varying shear rate.

<p><b>A)</b> Time series of the amount of thrombin showing the the maximum and minimum of the data (blue) generated by varying the shear rate from 1-1500 (1/s) as well as thrombin curves generated with shear rates 1, 10, 100, 500, and 1500 (1/s). Dependence of <b>B)</b> lag time*; <b>C)</b> maximum relative rate*; <b>D)</b> final amount on shear rate.</p

Sensitivity of thrombin generation to KRCs.

<p>Variation in the <b>A,D)</b> lag time; <b>B,E)</b> maximum relative rate; <b>C,F)</b> final concentration; to platelet characteristics using the local (OAT) method (<b>A-C</b>) and global Sobol method (<b>D-F</b>). <i>Local</i>: Sensitivities that lie between 0.75 and 1 (blue), between 0.25 and 0.75 (magenta), less than 0.25 (cyan) determine the rank-ordered list of kinetic rate constants. The percent change of thrombin generation measures from standard model output for each initial condition are represented by triangles. The direction of variation is indicated with an upwards or downwards facing triangle. <i>Global</i>: First and Total Order Sobol indices are plotted as bars with errors of 2 standard deviation about the mean, computed with 5,000 bootstrap samples of the original 540,000 function evaluations. The coefficient of variation is included to provide a scale for the fraction of variance. PCs with Total Order index statistically significantly larger than the First order index are indicated with a star.</p

Sensitivity of total thrombin generation to plasma levels.

<p>Variation in the <b>A,D)</b> lag time; <b>B,E)</b> maximum relative rate; <b>C,F)</b> final concentration; to zymogen and chemical inhibitor levels using the local (OAT) method (<b>A-C</b>) and global Sobol method (<b>D-F</b>). <i>Local</i>: Sensitivities that lie between 0.75 and 1 (blue), between 0.25 and 0.75 (magenta), less than 0.25 (cyan) determine the rank-ordered list of initial levels. The percent change of thrombin generation measures from baseline model output for each initial condition are represented by triangles. The direction of variation of the input parameter is indicated with an upwards or downwards facing triangle. <i>Global</i>: First and Total Order Sobol indices are plotted as bars with errors of 2 standard deviation about the mean, computed with 5,000 bootstrap samples of the original 110,000 function evaluations. The coefficient of variation is included to provide a scale for the fraction of variance. No total order index was statistically significantly larger than the first order index, indicating that the model output is not significantly effected by interactions between the parameters considered here.</p

A local and global sensitivity analysis of a mathematical model of coagulation and platelet deposition under flow

<div><p>The hemostatic response involves blood coagulation and platelet aggregation to stop blood loss from an injured blood vessel. The complexity of these processes make it difficult to intuit the overall hemostatic response without quantitative methods. Mathematical models aim to address this challenge but are often accompanied by numerous parameters choices and thus need to be analyzed for sensitivity to such choices. Here we use local and global sensitivity analyses to study a model of coagulation and platelet deposition under flow. To relate with clinical assays, we measured the sensitivity of three specific thrombin metrics: lag time, maximum relative rate of generation, and final concentration after 20 minutes. In addition, we varied parameters of three different classes: plasma protein levels, kinetic rate constants, and platelet characteristics. In terms of an overall ranking of the model’s sensitivities, we found that the local and global methods provided similar information. Our local analysis, in agreement with previous findings, shows that varying parameters within 50-150% of baseline values, in a one-at-a-time (OAT) fashion, always leads to significant thrombin generation in 20 minutes. Our global analysis gave a different and novel result highlighting groups of parameters, still varying within the normal 50-150%, that produced little or no thrombin in 20 minutes. Variations in either plasma levels or platelet characteristics, using either OAT or simultaneous variations, always led to strong thrombin production and overall, relatively low output variance. Simultaneous variation in kinetics rate constants or in a subset of all three parameter classes led to the highest overall output variance, incorporating instances with little to no thrombin production. The global analysis revealed multiple parameter interactions in the lag time and final concentration leading to relatively high variance; high variance was also observed in the thrombin generation rate, but parameters attributed to that variance acted independently and additively.</p></div

Schematic of thrombin metrics.

<p>Three physiologically relevant metrics of Thrombin generation were calculated for sensitivity analysis: 1) lag time (<i>Red</i>), 2) maximum relative rate of thrombin generation, measured after 0.1nM of thrombin have been obtained (<i>Green</i>), and 3) final concentration, the total concentration of thrombin at 20 minutes (<i>Blue</i>).</p

Variation in thrombin generation as a result of varying a subset of all model parameters.

<p><b>A)</b><i>Left Axis:</i> Thrombin concentration time series showing the mean (solid black line) and boundaries that encompass 50% of the data (purple), 90% of the data (orange), and the maximum/minimum of the computed solutions (gray-dashed) generated by uniformly varying a subset of all parameters (found via the Morris method) sampled between 50-150% of their nominal value simultaneously (740,000 total function evaluations). <i>Right Axis:</i> Marginal histogram of final thrombin concentration at <i>t</i> = 1200 seconds. <b>B)</b> Heatmap and marginal histograms relating three important thrombin generation metrics: lag time (<i>y-axis</i>), maximum relative rate (<i>x-axis</i>), and final concentration (<i>color-axis</i>). Results obtained by post-processing samples used to compute the global sensitivity indices. Dashed black bar in (A) and (B) represents the baseline case of 275nM of thrombin at 20 minutes.</p

Variation in thrombin generation as a result of varying plasma levels.

<p><b>A)</b><i>Left Axis:</i> Thrombin concentration time series showing the mean (solid black line) and boundaries that encompass 50% of the data (pink), 90% of the data (orange), and the maximum/minimum of the computed solutions (gray-dashed) generated by uniformly varying initial zymogen plasma levels from 50-150% of normal simultaneously (110,000 total function evaluations). <i>Right Axis:</i> Marginal histogram of final thrombin concentration at <i>t</i> = 1200 seconds. <b>B)</b> Heatmap and marginal histograms relating three important thrombin generation metrics: lag time (<i>y-axis</i>), maximum relative rate (<i>x-axis</i>), and final concentration (<i>color-axis</i>). Results were obtained by post-processing samples used to compute the global sensitivity indices. Dashed black bar in (A) and (B) represents the baseline case of 275nM of thrombin at 20 minutes.</p