Modeling Radiotrophic Fitness — Simulation Suite

Kinder & Faulkner (2026) · Systems Biology · bioRxiv preprint

A multi-scale simulation framework for radiotrophic biofilm communities in cylindrical bioreactors, spanning Langevin PSDE dynamics, 3D Cellular Potts lattice modeling, and a radiodialysis membrane transport PDE system for nuclear bioremediation design.

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Repository Structure

Biofilms/
├── biofilms.R                    # Flat-domain Langevin PSDE + k-means (original)
├── biofilms_3d.R                 # Cylindrical bioreactor — Shiny interactive app
├── biofilms_potts.jl             # 3D Cellular Potts Model + radiodialysis coupling
├── biofilms_radiodialysis.R      # Radiodialysis PDE system — Shiny interactive app
├── reactor_decision_tree.R       # Hamiltonian kNN reactor selection
├── preprint/
│   ├── modeling_radiotrophic_fitness.pdf   # 21-page preprint
│   ├── modeling_radiotrophic_fitness.tex   # LaTeX source
│   └── figures/                            # CairoMakie simulation figures (PDF + PNG)
└── assets/                                 # README preview images

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Mathematical Framework

The fitness of each species $s$ is governed by a PSDE coupling diffusion, radiation, melanin-mediated energy transduction, and Hamiltonian inter-species forces:

$$ \partial_t F_s = \nabla\cdot(D_s \nabla F_s) - \nabla\cdot!\Bigl(\mu_s \sum_j P_{sj}(t),F_j\Bigr) + R_s + \sigma_s \xi(t,\mathbf{x}) - \beta_{s,\text{ion}},I_\gamma F_s + \gamma_s \Delta_s - \alpha_{s,\text{nir}},N F_s + \theta_s H_s + C_s $$

The total multi-species Hamiltonian:

$$ H = \sum_i \Bigl[\tfrac{1}{2}\rho_i v_i^2 + U_i(\mathbf{x}_i)\Bigr] + \sum_{i \neq j} V_{ij}(r_{ij}) + \sum_k W_k(t,\mathbf{x}) $$

Mutualistic pairwise interaction potential:

$$ V_{ij}^{\text{mutual}} = -\gamma \exp!\Bigl(-\tfrac{r_{ij}^2}{\sigma^2}\Bigr) $$

Radiation field (Beer–Lambert, cylindrical source):

$$ I_\gamma(r) = I_0,e^{-\kappa r} $$

Melanin reaction-diffusion (radiotrophic fungi):

$$ \frac{\partial M}{\partial t} = D_M \nabla^2 M + \alpha_M \cdot n_\text{RF}(\mathbf{x},t) \cdot I_\gamma(t,\mathbf{x}) $$

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Radiodialysis Membrane Transport

Contaminant ingress through the bioreactor membrane under radiation-driven permeability change is modeled by a coupled three-equation system (§3.9 of the preprint):

Mobile contaminant (cylindrical reaction-diffusion):

$$\frac{\partial c}{\partial t} = \frac{1}{r}\frac{\partial}{\partial r}!\Bigl(r,D_\text{eff}\frac{\partial c}{\partial r}\Bigr) {-} (k_\text{ads}X {+} k_\text{red}X_\text{red}),c + k_\text{des},s$$

Immobile phase (biosorption + bioreduction):

$$\frac{\partial s}{\partial t} = (k_\text{ads}X + k_\text{red}X_\text{red}),c - (k_\text{des} + k_\text{loss}),s$$

Membrane damage and radiation-driven permeability:

$$\frac{dm}{dt} = -k_\text{dam},\dot{D}(R),m, \qquad P_\text{eff}(t) = P_0,\exp!\bigl(\alpha_P,D_\text{cum}(t)\bigr)$$

Robin boundary condition at the membrane wall r = R:

$$-D_\text{eff}\left.\frac{\partial c}{\partial r}\right|_{r=R} = P_\text{eff}(t),\bigl(c(R,t) - c_\text{ext}\bigr)$$

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Simulations

1 · Langevin PSDE — biofilms.R

Original flat-domain simulation. Seven species evolve under species-specific motility, radiation sensitivity, and pairwise Hamiltonian interactions. Stochastic Langevin integration with k-means spatial clustering.

 

Left: k-means cluster trajectories over 500 time steps. Right: full 7-species fitness field dynamics.

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2 · Cylindrical Bioreactor — biofilms_3d.R

Interactive Shiny app. Langevin dynamics inside a cylindrical bioreactor of radius R, axial length L. Radiotrophic species (C. neoformans, C. sphaerospermum) are attracted toward the high-radiation central axis; radiosensitive species drift outward. Sliders: radiation intensity I₀, attenuation κ, nutrient C₀, thorium intensity.

shiny::runApp("biofilms_3d.R")

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3 · Cellular Potts Model — biofilms_potts.jl

Pure Julia. 60³ cylindrical lattice, Metropolis MC, 5-term Hamiltonian (adhesion + volume + radiation + melanin + pairwise), coupled melanin/nutrient/radiation fields. Runs the radiodialysis PDE coupling by default; pass --no-radiolysis for plain CPM.

julia biofilms_potts.jl              # coupled CPM + radiodialysis
julia biofilms_potts.jl --no-radiolysis  # CPM only

Fig 1 — Radial stratification over 100 MCS. Radiotrophic fungi migrate toward the outer wall; B. subtilis retreats to the low-radiation core. Spatial sorting is emergent from the CPM Metropolis dynamics, not imposed.

Fig 2 — Melanin accumulation. C. sphaerospermum accumulates the most melanin (field value 1.44 at MCS 100) due to its radiotrophic positioning in the high-radiation outer zone. Melanin growth is linear over the simulation window — saturation not yet reached.

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4 · Radiodialysis Membrane Transport — biofilms_radiodialysis.R

Interactive Shiny app. Method-of-lines finite-volume solver for the three-equation radiodialysis PDE system. LSODA adaptive stiff integration (deSolve). Four visualization tabs: c(r,t) heatmap, s(r,t) heatmap, membrane integrity / P_eff time series, radial snapshots.

shiny::runApp("biofilms_radiodialysis.R")
# or headless:
Rscript biofilms_radiodialysis.R

Fig 3 — Membrane damage and permeability. Integrity m(t) decays exponentially under 50 Gy cumulative dose (1.0 → 0.78). P_eff rises 2.7× — the same radiation field that sustains the biofilm also opens the membrane wider, creating a self-regulating contaminant uptake loop.

Fig 4 — Contaminant penetration. Wall concentration c(R,t) reaches 87% of c_ext while the interior mean stays near zero — the biofilm consumes the contaminant within a thin annular zone at the membrane face. The slowly rising sorbed phase s_mean confirms progressive immobilisation.

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Key Results (from preprint §5)

Result	Value
Membrane integrity at 50 Gy	m = 0.779
P_eff increase	2.7× baseline (0.010 → 0.027 cm s⁻¹)
Interior contaminant mean	0.024 c_ext (98% depletion)
Radiotrophic stratification	C. neoformans mean r/R = 0.65; B. subtilis mean r/R = 0.50
C. sphaerospermum melanin (MCS 100)	1.44 (field units)
Pairwise community energy	−34.4 → −41.5 (tightening cooperation)
All species surviving	42 / 42 cells (no extinctions)

The central finding is a self-regulating remediation loop: radiation damages the membrane → P_eff increases → more contaminant enters → metal-reducing S. oneidensis (co-located at the outer wall by CPM dynamics) immobilises it. No external energy input required.

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Preprint

Modeling Radiotrophic Fitness — Kinder & Faulkner (2026) 21 pages · Systems Biology · Hamiltonian-Langevin framework · Cellular Potts Model · Radiodialysis PDE

Key sections:

• §3 Mathematical Framework (PSDE, Hamiltonian, radiation fields, melanin RD, radiodialysis)
• §4 Parameter Estimation (Table 2 — 7 species × 8 parameters, literature-justified)
• §5 Results (species clustering, CPM stratification, membrane transport, contaminant penetration)
• §6 Discussion (bioremediation implications, self-regulating loop design principle)

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Dependencies

R (≥ 4.2)

install.packages(c("deSolve", "shiny", "plotly", "ggplot2", "dplyr", "gridExtra"))

Julia (≥ 1.10)

import Pkg
Pkg.add(["CairoMakie", "GLMakie"])   # CairoMakie for export, GLMakie for interactive

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Citation

Kinder, H., Faulkner, B. (2026). Modeling Radiotrophic Fitness:
A Hamiltonian-Langevin Framework for Multispecies Biofilm Communities
under Ionising Radiation. bioRxiv preprint.