Reduced order modeling and analysis of the human complement system

<div><p>Complement is an important pathway in innate immunity, inflammation, and many disease processes. However, despite its importance, there are few validated mathematical models of complement activation. In this study, we developed an ensemble of experimentally validated reduced order complement models. We combined ordinary differential equations with logical rules to produce a compact yet predictive model of complement activation. The model, which described the lectin and alternative pathways, was an order of magnitude smaller than comparable models in the literature. We estimated an ensemble of model parameters from <i>in vitro</i> dynamic measurements of the C3a and C5a complement proteins. Subsequently, we validated the model on unseen C3a and C5a measurements not used for model training. Despite its small size, the model was surprisingly predictive. Global sensitivity and robustness analysis suggested complement was robust to any single therapeutic intervention. Only the simultaneous knockdown of both C3 and C5 consistently reduced C3a and C5a formation from all pathways. Taken together, we developed a validated mathematical model of complement activation that was computationally inexpensive, and could easily be incorporated into pre-existing or new pharmacokinetic models of immune system function. The model described experimental data, and predicted the need for multiple points of therapeutic intervention to fully disrupt complement activation.</p></div

Pairwise sensitivity and clustering of complement model parameters in the presence of 1 mg/ml zymosan.

<p>The response of the complement model was calculated for each parameter combination following a 10% increase in parameter combinations in the presence of 1 mg/ml zymosan. The model parameters were clustered into high (blue), medium (red) and low (green) response clusters based upon the euclidian distance between the perturbed and nominal system state (no perturbation).</p

Robustness analysis of the complement model.

<p>Robustness coefficients were calculated for a 50%, 90% and 99% reduction in C3, C5, or C3 and C5 initial conditions. <b>A</b>: Mean robustness index for C3a and C5a generated in the absence of zymosan. <b>B</b>: Mean robustness index for C3a and C5a generated in the presence of 1 mg/ml zymosan. The color describes the degree of reduction of C3a or C5a following the network perturbation. Robustness coefficients were calculated using all parameter sets with Pareto rank less than five (N = 65). Mean robustness values were reported.</p

Reduced order complement model predictions.

<p>Simulations of C3a and C5a generated using 0.1 mg/ml, 0.01 mg/ml, and 0.001 mg/ml zymosan were compared with the corresponding experimental measurements. The solid black lines show the simulated mean value of C3a or C5a for the ensemble, while the dark shaded region denotes the 99% confidence interval of mean. The light shaded region denotes the 99% confidence interval of the simulated C3a and C5a concentration. The experimental validation data (points) was taken from Morad et al [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0187373#pone.0187373.ref035" target="_blank">35</a>]. All initial conditions not specified by the experimental condition were assumed to be at zero or their physiological serum levels unless otherwise noted (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0187373#pone.0187373.s001" target="_blank">S1 Table</a>).</p

Reduced order complement model training.

<p>An ensemble of model parameters was estimated using multiobjective optimization from dynamic C3a and C5a measurements with and without zymosan. The model was trained using C3a and C5a measurements in the absence of zymosan (<b>A</b>–<b>B</b>) or in the presence of 1 mg/ml zymosan (<b>C</b>–<b>D</b>). The solid black lines show the simulated mean value of C3a or C5a for the ensemble, while the dark shaded region denotes the 99% confidence interval of mean. The light shaded region denotes the 99% confidence interval of the simulated C3a and C5a concentration. The experimental training data (points) was taken from Morad et al [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0187373#pone.0187373.ref035" target="_blank">35</a>]. All initial conditions not specified by the experimental condition were assumed to be at zero or their physiological serum levels unless otherwise noted (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0187373#pone.0187373.s001" target="_blank">S1 Table</a>).</p

Global sensitivity analysis of the reduced order complement model.

<p>Sensitivity analysis was conducted on the two objectives used for model training. <b>A</b>: Sensitivity of the C3a and C5a residual w/o zymosan. <b>B</b>: Sensitivity of the C3a and C5a residual with 1 mg/ml zymosan. The bars denote the mean total sensitivity index for each parameter, while the error bars denote the 95% confidence interval. <b>C</b>: Pathways controlled by the sensitivity parameters. Bold black lines indicate the pathway involves one or more sensitive parameters, while the red lines show current therapeutics targets. Current complement therapeutics were taken from the review of Morgan and Harris [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0187373#pone.0187373.ref039" target="_blank">39</a>].</p

Simplified schematic of the human complement system.

<p>The complement cascade is activated through three pathways: the classical, the lectin, and the alternative pathways. Complement initiation results in the formation of classical or alternative C3 convertases, which amplify the initial complement response and signal to the adaptive immune system by cleaving C3 into C3a and C3b. The C3 convertases further react to form C5 convertases which catalyze the cleavage of the C5 complement protein to C5a and C5b. C5b is critical to the formation of the membrane attack complex (MAC), while C5a recruits an adaptive immune response.</p