Multi-Species Continuum Biofilm Model

• Multi-species continuum biofilm model is a framework that applies continuum mechanics and PDEs to simulate the spatiotemporal evolution of microbial communities.
• It employs a Hamiltonian formulation with symmetric interaction matrices, ensuring thermodynamic consistency and robust numerical performance.
• The model is used to analyze biofilm dynamics in engineering, medical, and environmental settings, providing insights into species coexistence and antibiotic resistance.

A multi-species continuum-based biofilm model describes the temporal and spatial evolution of microbial communities composed of multiple bacterial species and extracellular substances, formulated at the continuum scale. These models are built on coupled systems of partial differential equations (PDEs) or hybrid frameworks, capturing cell growth, interspecies interactions, nutrient and chemical transport, mechanics, and responses to environmental and external perturbations.

1. Mathematical and Thermodynamic Foundation

Contemporary multi-species continuum-based biofilm modeling is grounded in continuum mechanics and thermodynamic principles. A prominent formulation derives the governing equations via Hamilton’s principle of stationary action, guaranteeing thermodynamic consistency by construction (Klempt et al., 1 Sep 2025). The Hamiltonian $\mathcal{H}$ integrates the system’s potential energy (free energy density $\Psi$), holonomic constraints, and dissipated energy (dissipation potential): $$\mathcal{H} = \int_{t} (\mathcal{G} + \mathcal{C} + \mathcal{D})\, dt$$ where:

• $\mathcal{G}$: total potential energy (function of biofilm composition, nutrient, antibiotic fields)
• $\mathcal{C}$: constraint functional (e.g., $\sum_{l=0}^n \varphi_l - 1 = 0$ for material incompressibility, with $\varphi_0$ denoting vacant space)
• $\mathcal{D}$: dissipated energy (dissipation function dependent on the rates of internal variables)

For each species $i$ $(i = 1, \dots, n)$, two internal variables are introduced: the overall volume fraction $\varphi_i$ and the percentage of living cells $\psi_i$; the product $\bar{\varphi}_i = \varphi_i \psi_i$ gives the effective volume fraction of living bacteria.

The evolution equations for $\varphi_i$ and $\psi_i$ are derived from stationarity of $\mathcal{H}$, subject to the constraint, and include species-dependent viscosity parameters $\eta_i$, nutrient and antibiotic concentrations ($c^*, \alpha^*$), and a Lagrange multiplier $\gamma$. These coupled, nonlinear, and, in general, rate-dependent equations ensure that the first and second laws of thermodynamics are satisfied by construction.

2. Coupling and Novel Interaction Schemes

A distinguishing feature of this framework is the use of a symmetric constant interaction matrix $A$ to encode both intra- and interspecies effects on biofilm dynamics (Klempt et al., 1 Sep 2025). The free energy density is formulated as: $$\Psi = -\frac{1}{2}c^* \bar{\boldsymbol{\varphi}}^T A \bar{\boldsymbol{\varphi}} + \frac{1}{2}\alpha^* \boldsymbol{\psi}^T B\boldsymbol{\psi}$$ where:

• $\bar{\boldsymbol{\varphi}}$ is the vector of living-cell fractions across species,
• $A$ is the constant symmetric growth coefficient matrix (diagonals: intrinsic species growth; off-diagonals: interspecies interaction),
• $\alpha^*$ is the antibiotic concentration field,
• $B$ is a diagonal matrix of antibiotic sensitivities for each species.

Positive off-diagonal elements in $A$ model protocooperative interactions; negative entries describe competitive or antagonistic behavior (e.g., allelopathy, ammensalism).

This interaction scheme is computationally tractable—it avoids auxiliary interfacial variables required by agent-based or dual-species continuum models and accommodates arbitrary numbers of species, facilitating extension from pairwise to complex, community-scale interaction structures.

3. Incorporation of External Fields

External environmental factors, namely, nutrient availability and antibiotic application, directly modulate the energy density and thus enter the evolution equations as explicit time-varying fields ($c^*, \alpha^*$). The nutrient term couples quadratically to species growth/competition, while the antibiotic term contributes a convex (quadratic) penalty on the living fraction, parameterized by the sensitivity matrix $B$. This coupling enables simulation of temporal protocols relevant for reactors, batch exposure cycles, or fluctuating environmental inputs, and can emulate perturbations (nutrient pulses, antibiotic dosing) in silico with stable and physically consistent results.

4. Validation, Model Behavior, and Key Results

The model has been validated via implicit time-stepping (Newton-Raphson) for canonical test cases (Klempt et al., 1 Sep 2025):

• Two-species simulations show that, absent explicit cooperation, the species with higher intrinsic growth dominates when spatial constraints (resource exhaustion, volume exclusion) are enforced. If off-diagonal interspecies coefficients are included, stable coexistence or richer dynamics (cooperation, competition, or both) can be maintained.
• Varying the diagonal antibiotic sensitivity entries in $B$ leads to selection for antibiotic-resistant species, in line with empirical trends.
• Four-species simulations confirmed that the framework is numerically robust for larger communities and can capture transitions such as non-monotonic evolution of living fractions ($\psi_i$), abrupt collapse on resource exhaustion, and responses to abrupt external field changes.

Quantitative agreement is reported with observed biofilm behavior, for example in the nonlinearity of population dynamics and abrupt collective transitions upon constraint violation or strong perturbations.

5. Comparison with Other Continuum-Based Frameworks

Multi-species continuum modeling is active across several regimes and design philosophies:

• Models based on reaction-diffusion-advection systems, derived via mass conservation, have been used to describe spatially resolved population and chemical fields, including migration, attachment, detachment, and substrate diffusion (Frunzo, 2017, Luongo et al., 2021, Owkes et al., 2023).
• Hamiltonian-based models (Klempt et al., 1 Sep 2025) offer automatic thermodynamic consistency and rigorous coupling between mechanics, resource fields, and biological state variables.
• Hybrid models, combining continuum fluid flow with agent-based descriptions or immersed boundary techniques, are suited for spatially explicit simulations of biofilm deformation, mechanical disruption, or fine-grained spatial heterogeneity (Hammond et al., 2013, Sudarsan et al., 2015).

The distinguishing advantages of the Hamiltonian/matrix-coupling approach are:

• Computational parsimony (no need for $O(n^2)$ interaction variables),
• Ready extensibility to arbitrary numbers of species,
• Natural inclusion of external perturbations with minimal parameter overhead,
• Guaranteed thermodynamic consistency.

6. Applications, Implications, and Future Directions

The continuum, matrix-coupled multi-species framework is applicable to:

• Engineering and medical biofilms, where antibiotic resistance, spatial competition, and cooperation drive system performance (bioreactors, infection control),
• Scenario analysis in environmental biofilm settings (wastewater, bioremediation, industrial fouling),
• In silico experimental design, enabling efficient exploration of protocols and parameter regimes.

Its extensibility (arbitrary $n$, external fields), stability, and relatively low complexity make it suitable for practitioners, including those less versed in advanced PDE modeling.

Future directions include direct spatial resolution (PDEs rather than material points), parameter identification/calibration with experimental multi-omics data, extension to structured microbial communities (e.g., compartmentalization), and integration with models addressing the transport and mechanics of the biofilm matrix (poroelasticity, non-Newtonian rheology, heterogeneity induced by EPS gradients).

7. Summary Table: Core Structures and Concepts

Feature	Model Expression or Method	Purpose / Function
Hamiltonian formulation	$$\mathcal{H} = \int (\mathcal{G} + \mathcal{C} + \mathcal{D}) dt$$	Guarantees thermodynamic consistency
State variables (species)	$(\varphi_i, \psi_i)$	Models volume fraction and living ratio
Species coupling	Matrix $A$, quadratic in $\bar{\boldsymbol{\varphi}}$	Efficient multi-species interaction scheme
External fields	$c^*$ (nutrient), $\alpha^*$ (antibiotic), matrix $B$	Direct environmental control
Constraint enforcement	Lagrange multiplier $\gamma$ for $\sum \varphi_i + \varphi_0 = 1$	Volume/space constraint
Numerical solution	Implicit time stepping (Newton’s method)	Robust, mass-conserving integration

This structure facilitates systematic investigation of complex microbial communities and their adaptive responses under variable ecological and pharmacological scenarios, within a mathematically rigorous and computationally accessible framework.