Bacillus anthracis, the causative agent of anthrax, produces anthrax toxin composed of protective antigen (PA), edema factor (EF), and lethal factor (LF). PA is essential for toxin entry into host cells and is a key immunogen in anthrax vaccines. Mechanistic mathematical models can improve understanding of infection mechanisms, as demonstrated by studies on Francisella tularensis. However, quantitative data on B. anthracis toxin dynamics is scarce, hindering the development of comprehensive within-host models. This study addresses this gap by using three independent in vitro datasets to build a quantitative mathematical description of PA production and degradation. This focus on PA is justified by its crucial role in toxin action and its targeting by FDA-approved treatments. Previous studies by Zai et al. and Charlton et al. investigated PA and LF production, showing sigmoidal growth curves and PA levels peaking then declining, possibly due to glucose depletion or protease activity. This study uses a new experimental dataset from the Defence Science and Technology Laboratory (Dstl) alongside the data from Zai et al. and Charlton et al. A key difference in our experimental approach is that we did not heat-activate B. anthracis spores before inoculation, allowing for observation of more natural germination dynamics and enabling calibration of additional model parameters.

Existing literature highlighted the importance of anthrax toxin, particularly PA, in the pathogenesis of anthrax. Studies by Zai et al. and Charlton et al. provided valuable in vitro data on PA production and bacterial growth kinetics. Zai et al. observed a peak and subsequent decline in PA levels, potentially due to glucose depletion or protease activity, while Charlton et al. observed higher PA levels and a different decay profile. These discrepancies in experimental findings and the overall scarcity of quantitative data on B. anthracis toxin dynamics motivated the current study. Previous modeling efforts, such as Day et al.'s work, included toxin dynamics but only qualitatively, lacking units and quantitative descriptions. The lack of a robust quantitative model incorporating PA production and decay underscored the need for this research.

The study employed a system of delay differential equations (DDEs) to model the in vitro dynamics of PA production. The model incorporates spore germination, bacterial maturation, logistic bacterial growth, PA production by vegetative bacteria, and PA decay (natural and protease-mediated). Time delays account for the lag in PA and protease production. Nutrient depletion (e.g., glucose) affects the PA production rate. The model was calibrated to three independent datasets (new Dstl data, Zai et al., and Charlton et al.) using Approximate Bayesian Computation Sequential Monte Carlo (ABC-SMC). ABC-SMC involves iterative sampling of parameter values, model simulation, comparison of model output to experimental data (using distance functions), and acceptance of parameter sets based on distance thresholds. Uniform priors were used for parameters (log-transformed in some cases). Gaussian noise was added to simulated data to account for measurement errors. Distance functions were carefully defined to handle different data types (bacterial CFU and PA concentrations) with different units. The Dstl experiment used B. anthracis Sterne strain 34F2. Spores were inoculated into BHI broth with sodium bicarbonate and CO2. Viable counts were determined by plating serial dilutions. Spore and vegetative bacterial counts were differentiated by heat treatment. PA concentration was measured using an automated western blot system (Jess™ Simple Western). The Zai et al. data used the A16R and Sterne strains, with PA quantified by ELISA. The Charlton et al. data simulated the UK vaccine manufacturing process using the Sterne strain, with PA measured by ELISA. Key differences between the experimental methods (e.g., spore heat activation, culture media, incubation methods) are discussed.

The DDE model successfully captured the dynamics observed in all three datasets. Parameter estimates from different datasets showed considerable consistency for many parameters, though some differences emerged. The Dstl experiment showed faster bacterial growth and a higher carrying capacity compared to Zai et al. and Charlton et al., possibly due to the richer BHI medium used. The Charlton et al. experiment exhibited a much higher PA yield, possibly attributed to static incubation. The model distinguished between natural PA decay (ν₀) and protease-mediated decay (ν). The Zai et al. data (A16R strain, with protease inhibitors) allowed for accurate estimation of ν₀, which was then fixed for other datasets. Analysis of datasets without protease inhibitor data revealed two parameter regimes explaining PA decay: (1) rapid nutrient depletion leading to cessation of PA production and subsequent decay; (2) slow nutrient depletion requiring higher protease-mediated decay rates to explain observed PA decline. Preliminary findings suggest nutrient depletion as the primary mechanism for PA decay in most datasets. For the Sterne and A16R strains from Zai et al., Sterne showed faster bacterial division and higher PA concentration. Parameter estimates from the Dstl data showed high consistency with those from Zai et al. (Sterne), although faster bacterial growth rate (λ) and higher carrying capacity (K) were observed. The delays in PA and protease production (τ₁ and τ₂) were estimated to be slightly longer in the Dstl experiment compared to Zai et al., possibly due to direct spore inoculation without pre-growth and the absence of heat activation.

The study successfully developed a DDE model that effectively described the in vitro dynamics of B. anthracis growth and PA production and decay across diverse experimental conditions and strains. The model's ability to distinguish between natural and protease-mediated PA decay provides valuable insights into the mechanisms governing PA levels. The consistency of parameter estimates across datasets enhances confidence in the model's robustness. Differences observed between datasets highlight the influence of experimental factors like culture media and incubation methods on bacterial growth and PA production. The identification of two parameter regimes for PA decay underscores the need for further research to elucidate the exact mechanisms. Future work could incorporate protease concentration data, explore in vivo PA dynamics, and refine the nutrient-PA production relationship. The quantitative understanding of PA dynamics provided by this model is a critical step towards developing comprehensive within-host models of anthrax infection, which could be used to inform treatment strategies.

This study provides a robust mathematical model for quantifying in vitro B. anthracis growth and PA production and decay, utilizing three independent datasets. The model successfully distinguishes between natural and protease-mediated PA decay, offering valuable insights into the underlying mechanisms. The consistency across datasets enhances model validity, although some differences highlight experimental nuances. Future research should focus on incorporating protease and nutrient data for more comprehensive model refinement and exploring in vivo implications. This work forms a foundation for future within-host models incorporating treatment strategies.

The model currently does not explicitly model protease concentration, relying on implicit representation through vegetative bacterial counts and time delays. The nutrient level is normalized, potentially obscuring differences in nutrient types and concentrations across experiments. The assumed proportionality between nutrient level and PA production rate might be an oversimplification. Extrapolation of in vitro findings to in vivo settings requires caution, as host-mediated PA degradation and other factors are not explicitly included.