PlasmaCellDiff

Codebase implementing the germinal center exit pathway kinetic model, emphasizing bifurcation analysis through simulation, as introduced by Martinez et. al. (2012). Part of the masters course Bioinformatics 1 during period 1, 2023 at the University of Amsterdam. A summary of findings is contained in frazier_jared_bioinformatics_report.pdf.

Installation

The only requirement is Julia, preferably version 1.9. Clone the repo and start a Julia REPL in the root directory

julia --project=path/to/PlasmaCellDiff

press ] to enter Pkg mode. The REPL prompt should like

(PlasmaCellDiff) pkg>

Then instantiate the package to download the appropriate dependencies by

(PlasmaCellDiff) pkg> instantiate

Should you wish to re-run the notebooks that generate the figures, hit backspace to return to the regular REPL

julia> using IJulia
julia> notebook(dir="notebooks/")

then you can open notebooks and run them accordingly to reproduce the figures in the figures/ directory.

Germinal Center Model

Note that $bcr_0$ and $cd_0$ could either be modelled with the Gaussian PDF defined below, or they could simply be constant parameters of the model.

$$ \begin{aligned} \frac{dp}{dt} &= \mu_p + \sigma_p \frac{k_b^2}{k_b^2 + b^2} + \sigma_p \frac{r^2}{k_r^2 + r^2} - \lambda_p p \\ % BLIMP1 \frac{db}{dt} &= \mu_b + \sigma_b \frac{k_p^2}{k_p^2 + p^2}\frac{k_b^2}{k_b^2 + b^2}\frac{k_r^2}{k_r^2 + r^2} - (\lambda_b + BCR)b \\ % BCL6 \frac{dr}{dt} &= \mu_r + \sigma_r \frac{r^2}{k_r^2 + r^2} + CD40 - \lambda_r r\\ % IRF4 BCR &= bcr_0 \frac{k_b^2}{k_b^2 + b^2} \\ % BCR CD40 &= cd_0 \frac{k_b^2}{k_b^2 + b^2} \\ % CD40 bcr_0 &= BCR_{max} \frac{1}{\sigma_{BCR}\sqrt{2\pi}}\exp[-\frac{1}{2}(\frac{t - \mu_{BCR}}{\sigma_{BCR}})^2] \\ % Gaussian bcr0 cd_0 &= CD40_{max}\frac{1}{\sigma_{CD40}\sqrt{2\pi}}\exp[-\frac{1}{2}(\frac{t - \mu_{CD40}}{\sigma_{CD40}})^2] % Gaussian cd0 \end{aligned} $$

References

M. R. Martínez et al., Proceedings of the National Academy of Sciences, vol. 109, no. 7, pp. 2672–2677, 2012. doi:10.1073/pnas.1113019109

Good Example of Scientific Project in Julia: HighDimensionalComplexityEntropy

Datseris G. and Parlitz U. Chapter 1 - 4 from "Nonlinear Dynamics: A Concise Introduction Interlaced with Code"(2022). github. note: null clines, bifurcation analysis, stability analysis, lyapunov exponents.

Numerical bifurcation diagrams and bistable systems. Lecture notes from computational biology at École normale supérieure de Paris. url: http://www.normalesup.org/~doulcier/teaching/modeling/ (2018)