# A multiscale coupled PDE model of host-microbiota interaction: application to colonic mucosal barrier dynamics

Eleonora Pastremoli, Béatrice Laroche, Lorenzo Sala

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Eleonora Pastremoli, Béatrice Laroche, Lorenzo Sala. A multiscale coupled PDE model of host-microbiota interaction: application to colonic mucosal barrier dynamics. 2026. hal-05524422

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# A MULTISCALE COUPLED PDE MODEL OF HOST-MICROBIOTA INTERACTION: APPLICATION TO COLONIC MUCOSAL BARRIER DYNAMICS *

ELEONORA PASTREMOLI†‡, BÉATRICE LAROCHE†‡, AND LORENZO SALA†‡

**Abstract.**

This study presents a multiscale mathematical model based on a system of coupled Partial Differential Equations (PDEs) that integrates microbial dynamics with the host's colonic structure, emphasizing the accurate modeling of the mucosal barrier effect. This coupled colon-crypt model is designed to investigate complex interactions between the gut microbiota and host defense mechanisms under healthy and perturbed conditions. Validation against previous models and literature data confirms that the model accurately captures the dynamics of the intestinal crypt, cellular composition, and the distinct structural behaviors of the inner (hard) and outer (soft) mucus layers. This work provides a mathematically grounded and computationally tractable platform for studying colonic mucosal function and its alteration under pathological conditions, and constitutes a step towards patient-specific *in silico* modeling.

**Key words.** Advection-Diffusion-Reaction system, Host-Microbiota Interaction, PDE Model, Multiscale Modeling, Mucosal Barrier

**1 Introduction** The gut microbiota is a dynamic ecosystem essential for nutrient absorption, immune modulation, and metabolic regulation. Dysbiosis, a disruption in its composition or function, is strongly associated with inflammatory and metabolic disorders. The colon, especially significant in mediating host-microbiota interactions, consists of two key compartments: the lumen, which harbors the microbiota, and the mucosa, organized into intestinal crypts responsible for mucus secretion and nutrient absorption. As shown in Figure 1, mucus secreted by goblet cells (GCs) forms a stratified barrier: a dense hard inner layer physically excludes bacteria from the epithelium, and an outer soft layer more accessible to microbial colonization (11; 18).

<figure><img src="image_2.png" /><figcaption>Fig. 1: **Mucus architecture in a colonic crypt.** The left panel highlights the hard (red) and soft (light blue) mucus layers, the location of goblet cells (GCs), and the lumen containing microbiota. The right panel provides a schematic representation of the main flows in the model (Figure adapted from (18)).</figcaption></figure>

While existing computational models have explored gut physiology, they often

<footnote>†Université Paris-Saclay, INRAE, MaIAGE, 78350 Jouy-en-Josas, France.</footnote>

<footnote>‡Université Paris-Saclay, INRIA, MUSCA, 91120 Palaiseau, France.</footnote>

<footnote>*Submitted to the editors February 23, 2026.</footnote>
lack an integrative multiscale perspective. Simplified 1D models have shown that flow velocity and peristaltic mixing determine bacterial density profiles (2; 6), yet they omit the radial complexity of mucus and crypts. Similarly, while some studies addressed the optimization of microstructures for nutrient absorption (9) or the impact of flow on diversity (16), they often rely on purely diffusive transport or simplified geometries, neglecting the convective contributions and the biological boundary conditions of the mucosal surface. To bridge these gaps, our group has previously developed models for macroscopic fluid-microbiota interactions (15), microscopic crypt morphology (7; 10), and organ-scale metabolic reactions (17). Building on these foundations, this study introduces a multiscale computational framework that integrates microbial dynamics with the colonic mucosal structure, specifically modeling the dual-layer mucus system. This framework spans from cellular processes within crypts to tissue-level dynamics, providing a unified representation of host-microbiota interactions across scales. In the future, by incorporating patient-specific data such as microbiota composition and diet, the model aims to predict clinical outcomes and optimize personalized treatment strategies.

The paper is organized as follows: Section 2 details the mathematical model, numerical methods and computational challenges; Section 3 presents model validation and results; finally, Section 4 summarizes the conclusions and future directions.

**2 Mathematical model** The proposed mathematical formulation builds upon existing Partial Differential Equation (PDE)-based colon and crypt models (15; 10), spanning from the macroscopic dynamics of the colon to the microscopic scale of epithelial crypts. To improve biological accuracy, the colon model has been refined to explicitly distinguish between soft and hard mucus layers (Figure 1) and implement reformulated mucosal secretion and absorption mechanisms to ensure temporal continuity. Furthermore, the crypt model was extended to incorporate active cell migration as a primary driver of epithelial turnover, reflecting recent experimental evidence (14). Crucially, we introduced an original bidirectional coupling mechanism that enables a dynamic exchange of entities between the two scales, providing a more comprehensive framework for studying the integrated gut ecosystem.

**2.1 Geometrical Assumptions** The colon is modeled as a cylindrical domain $\Omega_{colon} = \omega \times [0, L] \subset \mathbb{R}^3$, where $\omega \subset \mathbb{R}^2$ is a disc of radius $R$ representing its transverse section. Its boundaries $\Gamma_{in} = \omega \times \{0\}$, $\Gamma_{out} = \omega \times \{L\}$ and $\Gamma_m = \partial\Omega_{colon} \setminus (\Gamma_{in} \cup \Gamma_{out})$ represent the proximal inlet, the endpoint of the modeled portion, and the mucosal wall, respectively. Assuming cylindrical symmetry, the problem is reduced to a 2D domain $(r, z)$, with $r \in [0, R]$, highlighted in yellow in Figure 2a.

The crypts are modeled with a truncated test-tube geometry with cylindrical symmetry around its vertical axis $[0, Z_{max}]$. Neglecting radial phenomena compared to vertical dynamics (7), we employ a 1D spatial model where each cell is identified by its vertical position $z \in [0, Z_{max}]$ (Figure 2b). The two models are dynamically linked via bidirectional mass exchange at the mucosal interface $\Gamma_m$. To maintain computational efficiency despite the high physiological crypt density ($N_{crypt} \approx 10^4 \text{ cm}^{-2}$), we adopt an architecture based on three representative average crypts (Figure 2c). These crypts are distributed along the $L = 10 \text{ cm}$ colon length, representing colon segments of 2, 5, and 3 cm, respectively. Each representative crypt is scaled by $N_{crypt}$ to account for the total epithelial surface of its section, allowing for the simulation of spatial heterogeneity and localized perturbations.
<figure><img src="image_3.png" /><figcaption>Fig. 2: **Geometry of the computational domains.** Schematic representation of (a) the cylindrical colon domain, (b) the one-dimensional crypt spatial domain, and (c) the colon domain coupled with the three average crypts.</figcaption></figure>

**2.2 Colon model** Following (15), the colon model is derived from mixture theory (5; 19), well adapted to the multiphasic intestinal content. The different entities considered (see Table 1) are classified into solid components (indexed by set $I_S$), which are large enough to generate mechanical forces, and dissolved compounds (indexed by set $I_D$), which are distributed within the mixture without affecting its flow. To enhance biological realism, this model was extended by incorporating additional entities described in (10) and, above all, a newly identified hard mucus layer specifically designed to protect the epithelium by keeping bacteria at a distance.

<table><caption>Table 1: Classification of entities in the colon model and definition of $I_S$ and $I_D$.</caption><thead><tr><th colspan="2">Solid components</th><th colspan="2">Dissolved compounds</th></tr><tr><th>Name</th><th>Notation</th><th>Name</th><th>Notation</th></tr><tr><td colspan="2">Elements of set I<sub>S</sub></td><td colspan="2">Elements of set I<sub>D</sub></td></tr></thead><tbody><tr><td>Soft mucus</td><td>sm</td><td>Monosaccharides</td><td>mon</td></tr><tr><td>Hard mucus</td><td>hm</td><td>Lactate</td><td>la</td></tr><tr><td>Polysaccharides</td><td>pol</td><td>Acetate</td><td>ac</td></tr><tr><td>Proteins</td><td>prot</td><td>Propionate</td><td>pro</td></tr><tr><td>Bacterial groups</td><td>[bact]</td><td>Butyrate</td><td>but</td></tr><tr><td>Dietary residuals</td><td>r</td><td>Hydrogen sulfide</td><td>H<sub>2</sub>S</td></tr><tr><td>Liquid chyme</td><td>l</td><td>Methane</td><td>CH<sub>4</sub></td></tr><tr><td></td><td></td><td>Carbon dioxide</td><td>CO<sub>2</sub></td></tr><tr><td></td><td></td><td>Hydrogen</td><td>H<sub>2</sub></td></tr><tr><td></td><td></td><td>Oxygen</td><td>O<sub>2</sub></td></tr><tr><td></td><td></td><td>Antimicrobial peptides</td><td>AMPs</td></tr></tbody></table>

Bacterial solid components are defined following (10), comprising 8 functional microbial groups: $[bact] = \{B_{mon}^{\Phi}, B_{mon}^{\eta}, B_{la}^{\Phi}, B_{la}^{\eta}, B_{la}^{\delta}, B_{H2S}^{\delta}, B_{H2a}^{\Phi}, B_{H2m}^{\Phi}\}$. These groups are categorized by: (i) metabolic capacity, including monosaccharide and lactate utilizers ($mon, la$) and hydrogen consumers ($H2S, H2a, H2m$) producing hydrogen sulfide, acetate, and methane; and (ii) inflammatory response profiles, categorized by sensitivity to host inflammatory signals as high ($\Phi$), moderate ($\eta$), or low/pro-motivated ($\delta$). In this framework, oxygen concentration serves as the primary marker for inflammatory conditions (10). Detailed classifications and physiological processes are
provided in Appendix SM2.

It is assumed that the solid components are mainly composed of water and have the same volumetric mass. Hence, the solid components $i \in I_S$ are modeled through their volume fractions $f_i$, while the soluble compounds $j \in I_D$ through their concentrations $c_j$. Moreover, solid components are assumed to fill the entire available volume, so that:

$$ (2.1) \qquad \sum_{i \in I_S} f_i(x, t) = 1, \text{ for any } t > 0, x \in \Omega_{\text{colon}}. $$

The time evolution of $f_i$ and $c_j$ are governed by the following advection-diffusion-reaction (ADR) equations:

$$ (2.2) \qquad \partial_t f_i - \text{div}(\sigma \nabla f_i) + \text{div}(f_i u_i) = F_i \qquad \text{in } \Omega_{\text{colon}} $$

$$ (2.3) \qquad \partial_t c_j - \text{div}(\sigma_j \nabla c_j) + \text{div}(c_j \tilde{u}) = G_j \qquad \text{in } \Omega_{\text{colon}} $$

where $\tilde{u} = \sum_{i \in I_S} f_i u_i$ is the mixture velocity, $\sigma$ and $\sigma_j$ are the diffusion coefficients, and $F_i$ and $G_j$ are the source terms, representing the metabolic transformation rate. Importantly, we assume that the diffusion coefficient is the same for all solid phases and mass conservation is satisfied through the metabolic transformations, so that:

$$ (2.4) \qquad \sum_{i \in I_S} F_i(x, t) = 0, \text{ for any } t > 0, x \in \Omega_{\text{colon}}. $$

Note that summing Eq. (2.2) for all solid phases and applying the constraints in Eqs. (2.1)-(2.4) yields the following condition:

$$ (2.5) \qquad \text{div}(\tilde{u})(x, t) = 0, \text{ for any } t > 0, x \in \Omega_{\text{colon}}. $$

The velocity $u_i$ accounts for component-specific transport mechanisms and differs across solid components to reflect their distinct transport behaviors. It is defined as:

$$ (2.6) \qquad u_i = \begin{cases} u, & \text{if } i \in \{sm, l\} \\ [\frac{U_a}{0}] & \text{if } i = hm \\ u + [\frac{U_r}{0}] & \text{if } i \in I_S \setminus \{hm, sm, l\} \end{cases} $$

with $u$ the fluid velocity, governed by the Stokes equations for a non-Newtonian fluid:

$$ (2.7) \qquad \left\{ \begin{array}{l} -\nabla \cdot (\mu(f)\varepsilon(u)) + \nabla p = 0 \\ \nabla \cdot u = -\nabla \cdot \left( f_{hm} \left( [\frac{U_a}{0}] - u \right) + \left( \sum_{i \in I_S \setminus \{hm, sm, l\}} f_i \right) [\frac{U_r}{0}] \right) \end{array} \right. \qquad \begin{array}{l} \text{in } \Omega_{\text{colon}} \\ \text{in } \Omega_{\text{colon}} \end{array} $$

Here $\varepsilon(u) := \frac{1}{2}(\nabla u + \nabla u^\top)$, $\mu(f) = \max(\mu_m(f_{hm}(x, z, t)), \mu_l(f_l(x, z, t)))$ is an inhomogeneous viscosity term and $p$ is the pressure. The divergence condition is a direct consequence of Eqs. (2.5)-(2.6).

The primary contributors to the viscosity $\mu(f)$ are the hard mucus and the liquid volume fractions:

* $\mu_m(f_{hm})$ describes the mucus rheology and depends on the hard mucus volume fraction. It is defined as:

$$ (2.8) \qquad \mu_m(f_{hm}) = \mu_{m,\min} + (\mu_{m,\max} - \mu_{m,\min}) \frac{f_{hm}^{\alpha_m}}{f_{hm,\text{thr}}^{\alpha_m} + f_{hm}^{\alpha_m}} $$
where $\mu_{m,min}$ and $\mu_{m,max}$ represent the asymptotic viscosity limits for low and high hard mucus concentrations, respectively, $f_{hm,thr}$ is the activation threshold, serving as a marker for the hard mucus layer, and $\alpha_m$ is a shape parameter that tunes the steepness of the transition.

* $\mu_l(f_l)$ describes the luminal rheology and depends on the liquid volume fraction. It is defined as:

$$ (2.9) \qquad \mu_l(f_l) = \mu_{l,\max} - (\mu_{l,\max} - \mu_{l,\min}) \frac{f_l^{\alpha_l}}{f_{l,thr}^{\alpha_l} + f_l^{\alpha_l}} $$

where the parameters $\mu_{l,min}$, $\mu_{l,max}$, $f_{l,thr}$ and $\alpha_l$ carry the same physical interpretation as described for the mucus phase, but are calibrated to the rheological properties of the luminal liquid (15).

For solid components other than mucus and liquid, the extra radial velocity component $U_r$ models the unidirectional repelling effect of hard mucus. This mechanism prevents these entities from approaching the epithelial boundary, thereby reflecting its barrier function. The repulsion follows a sigmoidal dependence on the hard mucus volume fraction $f_{hm}$, as defined in Eq. (2.10). For the hard mucus component ($i = hm$), the fluid velocity $u$ from the Stokes equation is not included because this mucus layer is composed of filaments (Figure 1) that are strictly attached to the epithelium and, thus, are not transported (11; 18). Instead, an effective attractive radial force $U_a$ is introduced to confine the hard mucus to the epithelial surface. This interaction is formalized in Eq. (2.11), which enforces a sigmoidal dependence of $U_a$ on $f_{hm}$. The resulting radial velocity components $U_r$ and $U_a$ are defined as follows:

$$ (2.10) \qquad U_r = C_{bot} + \frac{C_{top} - C_{bot}}{1 + e^{-k(f_{hm}-thr)}} $$

$$ (2.11) \qquad U_a = A_{top} + \frac{A_{bot} - A_{top}}{1 + e^{-k(f_{hm}-thr)}} $$

where $C_{bot}$, $C_{top}$, $A_{bot}$, $A_{top}$ are the asymptotic constants representing the minimum and maximum values of the functions at the bottom and top of the transition, respectively, $k$ is the steepness parameter, which determines the slope of the sigmoidal transition, and $thr$ is the threshold value, delineating the critical volume fraction of hard mucus required to establish a functional barrier. Comprehensive details on these parameters, including their units and values used in the simulations, are summarized in Appendix Table SM1 and Appendix Table SM2.

The source terms $F_i$ and $G_j$ are derived from the framework proposed in (10), here extended to incorporate a comprehensive set of 14 metabolic and regulatory processes. Key additions include the degradation of hard mucus, along with the localized elimination of oxygen ($O_2$) and degradation of antimicrobial peptides (AMPs) during microbial metabolism and interaction. The remaining processes include the breakdown of dietary polysaccharides and mucus glycoproteins, the consumption of monosaccharides and lactate, and the production of short chain fatty acids and gases. Additionally, the production of hydrogen sulfide ($H_2S$) through dietary proteins is also included. Finally, the regulatory terms describe bacterial mortality, modulated by natural decay, AMPs concentration, and oxygen-driven inflammation sensitivity, as well as the toxic impact of $H_2S$ on mucus integrity. The derivation and the explicit mathematical definitions of these source terms are detailed in Appendix SM3 and Appendix SM4, respectively.

### 2.2.1 Boundary conditions
In order to model the transfers of mass across the system boundaries, boundary fluxes $\gamma_{f_i}$ and $\gamma_{c_j}$ are introduced. The mass conserva-
tion equations are therefore supplemented by the natural Robin boundary conditions:

$$ (2.12) \qquad (-\sigma \nabla f_i + f_i u_i) \cdot n = \gamma_{f_i} \qquad i \in I_S $$

$$ (2.13) \qquad (-\sigma_j \nabla c_j + c_j \tilde{u}) \cdot n = \gamma_{c_j} \qquad j \in I_D $$

where $n$ denotes the outward unit normal vector to the boundary. The specific expressions for the boundary fluxes $\gamma_{f_i}$ and $\gamma_{c_j}$ depend on the nature of the exchange, whether it is a constant influx, a spatially dependent function, or influenced by local phase volume fractions.

The dietary inflow on $\Gamma_{in}$, representing the daily volume of digestate reaching the proximal colon, is a key external input, serving as a primary nutrient source for the microbial community. In this region, the boundary fluxes are defined by the relations:

$$ (2.14) \qquad \gamma_{f_i} = u_{in} f_{i,in} \qquad \text{on } \Gamma_{in} $$

$$ (2.15) \qquad \gamma_{c_{mon}} = u_{in} c_{mon,in} \qquad \text{on } \Gamma_{in} $$

where $u_{in}$ denotes the prescribed inflow velocity profile, $f_{i,in}$ for $i \in I_S \setminus \{hm\}$ and $c_{mon,in}$ are the volume fractions and concentration, respectively, of the corresponding components in the inflow. Given that the proximal colon is being modeled, this represents the only inflow considered. All other model entities are subject to homogeneous Robin boundary conditions, as they are produced locally at later stages of the process.

The outflow on $\Gamma_{out}$ is defined to balance the net mass flux across the boundaries, ensuring conservation of the overall volume. In particular,

$$ (2.16) \qquad \gamma_{f_i} = f_i u \qquad \text{on } \Gamma_{out} $$

$$ (2.17) \qquad \gamma_{c_j} = c_j \tilde{u} \qquad \text{on } \Gamma_{out} $$

for all $i \in I_S \setminus \{hm\}$ and $j \in I_D$. In the specific case of hard mucus, the inflow and outflow boundary conditions are homogeneous Robin, as it is composed of filamentous structures that cannot be transported.

The Stokes system is endowed with the following boundary conditions:

$$ (2.18) \qquad \left\{ \begin{array}{ll} u \cdot n = -u_{in} & \text{on } \Gamma_{in} \\ u_r = \frac{1}{1-f_{hm}} \left[ \sum_{i \in I_S} \gamma_{f_i} - f_{hm} \begin{bmatrix} U_0 \\ 0 \end{bmatrix} \right] \cdot n & \text{on } \Gamma_m \\ -\left( \sum_{i \in I_S \setminus \{hm,sm,l\}} f_i \right) \begin{bmatrix} U_r \\ 0 \end{bmatrix} \cdot n + U_{per,r} & \text{on } \Gamma_m \\ u_z = U_{per,z} & \text{on } \Gamma_m \\ (-p\mathbb{I} + \mu(f)\varepsilon(u)) \cdot n = 0 & \text{on } \Gamma_{out} \end{array} \right. $$

By definition, the saturation constraint (Eq. (2.1)) implies that $f_{hm} < 1$ must hold for the fluid phase to exist. This condition is crucial for the mathematical consistency of the boundary condition on $\Gamma_m$.

Since the cylindrical computational domain is fixed and does not include moving boundaries, the effect of mucosal motility is modeled by prescribing a wall velocity profile $U_{per} = (U_{per,r}, 0, U_{per,z})$. It accounts for the net motile activity of the mucosal wall, including peristalsis. Here, $u_{in}$ is the prescribed inflow velocity profile. Parameter definitions, units, and values adopted for the simulations are reported in Appendix Table SM2.
**2.3 Crypt model** The intestinal crypt is regulated by complex processes of cell proliferation, differentiation, and migration, and is structured into distinct zones: the bottom hosts the stem cell niche, containing stem cells (sc) and deep crypt secretory cells (dcs); the middle contains progenitor cells (pc) undergoing division and differentiation; and the upper part consists of terminal cells, mainly enterocytes (ent) and goblet cells (gc). Cells produced at the bottom migrate upward and differentiate until they are removed by extrusion at the top. Microbiota-derived metabolites, primarily acetate (ac), propionate (pro), lactate (la), and butyrate (but), diffuse into the crypt. Together with blood-derived oxygen ($O_2$), these are metabolized by differentiated cells via β-oxidation, a key process for maintaining colonic anaerobiosis. Furthermore, enterocytes actively absorb these metabolites. If butyrate levels fall, cells switch from β-oxidation to glycolysis, a transition inhibited by the diffusion of $H_2S$.

The crypt model, originally introduced as a large population limit of a birth-death process (7) and later refined in (10), features a 1D spatial structure representing the vertical axis (Section 2.1). The system consists of two interacting components governed by coupled PDEs: the dynamics of the five-cell-type densities, $\rho = (\rho_s, \rho_p, \rho_{dcs}, \rho_g, \rho_e)^T$ with $\rho_{dcs}$ assumed to be constant in time, and the evolution of metabolite concentrations, $\tilde{c} = (\tilde{c}_{ac}, \tilde{c}_{la}, \tilde{c}_{pro}, \tilde{c}_{but}, \tilde{c}_{H_2S}, \tilde{c}_{O_2})^T$. Let $\bar{\rho}$ denote the total cell density along the crypt axis $[0, Z_{max}]$. For $l \neq dcs$, the dynamics of cell type $l$ are governed by:

$$ (2.19) \qquad \partial_t \rho_l + W \partial_z (\phi(z) (\partial_z \bar{\rho} + \alpha \bar{\rho}) \rho_l) = H_l(\rho_l, z, \bar{\rho}, \tilde{c}). $$

The reaction terms $H_l$, modeling cell fate and regulation, and the shape function $\phi$ remain consistent with (10). Both are $C_1$ functions of their arguments and a detailed description is provided in Appendix SM6. The transport term in Eq. 2.19 combines two mechanical effects: a passive, pressure-driven component $W \partial_z \bar{\rho}$ (10), and a novel active term $\alpha W \bar{\rho}$. The latter represents actin-mediated secretion proportional to local cell density, capturing active epithelial migration. Boundary conditions consist of Dirichlet at the crypt base and homogeneous Robin at the top (Appendix SM6).

Acetate, lactate, propionate, butyrate, and oxygen follow a reaction-diffusion equation along the crypt axis. The reaction term accounts for both β-oxidation and epithelial absorption. In contrast, hydrogen sulfide follows a standard diffusion equation. These formulations, largely consistent with (10), are detailed in Appendix SM7. At the crypt base ($z = 0$), boundary conditions include a Dirichlet condition for oxygen and homogeneous Neumann conditions for other metabolites. At the crypt top ($z = Z_{max}$), the conditions account for the multiscale coupling with the colon model, as detailed in the following section.

**2.4 Exchange through the mucosa** Boundary mass transfer is central to the model, mediating lumen–epithelium exchanges that drive both fluid dynamics and biochemical activity due to absorption and secretion of metabolites. Unlike previous formulations (15; 10), this model refines lumen–epithelium interactions by implementing a multiscale coupling between colon–level dynamics and the crypt model. This framework allows for a bidirectional information exchange between the host and microbiota. The coupling occurs via specific boundary fluxes imposed on the mucosal wall $\Gamma_m$ within the colon domain $\Omega_{colon}$. These fluxes, adapted from (15; 10), now account for the continuous secretion and absorption processes occurring within the crypt’s goblet cells and enterocytes.

Notably, metabolite absorption is not modeled as a simple linear function of liquid
concentration ($\gamma_{f_l} \cdot c_j$). Despite metabolites being dissolved in the liquid phase $f_l$, the presence of active transport mechanisms necessitates a more complex treatment than passive diffusion. Consequently, the crypt-to-colon exchanges are modeled as follows:

### Mucus production:

$$ (2.20) \qquad \gamma_{f_{sm}} = -\alpha_{sm} \cdot q_m \cdot N_{crypt} \cdot \overline{N_{gc}} \qquad \text{on } \Gamma_m $$

$$ (2.21) \qquad \gamma_{f_{hm}} = -\alpha_{hm} \cdot q_m \cdot N_{crypt} \cdot \overline{N_{gc}} \qquad \text{on } \Gamma_m $$

where $\overline{N_{gc}} = \int_{0}^{Z_{max}} \rho_g(z) dz$ is the total number of goblet cells per crypt, $q_m$ represents the mucus production rate per goblet cell and $N_{crypt}$ denotes the crypt density per surface unit. It is assumed that the production rate of total mucus is partitioned into two distinct constants: $\alpha_{sm}$ for soft mucus and $\alpha_{hm} = 1 - \alpha_{sm}$ for hard mucus. We specifically hypothesize that $\alpha_{sm} > \alpha_{hm}$, which implies that the quantity of soft mucus produced is proportionally greater than the quantity of hard mucus produced.

### Oxygen diffusion:

$$ (2.22) \qquad \gamma_{c_{O_2}} = -q_{O_2} \cdot \tilde{c}_{O_2}(Z_{max}, t) \cdot N_{crypt} \qquad \text{on } \Gamma_m $$

where $\tilde{c}_{O_2}(Z_{max}, t)$ is the oxygen concentration at the top of the crypt, given by the crypt model, and $q_{O_2}$ is the oxygen diffusion per crypt.

### Antimicrobial peptides diffusion:

$$ (2.23) \qquad \gamma_{c_{AMPs}} = -q_{AMPs} N_{crypt} \overline{N_{ent}} - q_{AMPs}^{imm} \mathbf{1}\{\int_{0}^{2\pi} \int_{R-d}^{R} \sum_{i \in [bact]} f_i(z,r) r dr d\theta > C_{bact}\} \qquad \text{on } \Gamma_m $$

where $\overline{N_{ent}} = \int_{0}^{Z_{max}} \rho_e(z) dz$ is the total number of enterocyte cells in one crypt, $q_{AMPs}$ is the absorption rate per enterocyte cell, $q_{AMPs}^{imm}$ represents the immune-induced production rate activated when bacteria penetrate the hard mucus layer, and $C_{bact}$ represents the bacterial threshold required to trigger the immune response on the mucosal wall. The parameter $d$ defines the width of an annular region adjacent to the boundary $\Gamma_m$. It is selected to ensure that the integration domain $[R - d, R]$ is entirely contained within the hard mucus layer. Specifically, $d$ is naturally related to the parameter $thr$ (introduced in Eqs. (2.10)–(2.11)), which denotes the critical hard mucus concentration required to establish a functional barrier.

Conversely, the colon-to-crypt exchanges include:

### Liquid absorption:

$$ (2.24) \qquad \gamma_{f_l} = q_l \cdot N_{crypt} \cdot f_l \cdot \overline{N_{ent}} \qquad \text{on } \Gamma_m $$

where $q_l$ is the water pumping per enterocyte cell.

### Dissolved components $j \in \{la, ac, pro, but, H_2S\}$:

$$ (2.25) \qquad \gamma_{c_j} = q_j \cdot N_{crypt} \cdot \overline{N_{ent}} \cdot \tilde{c}_j(Z_{max}, t) \qquad \text{on } \Gamma_m $$

$$ (2.26) \qquad \tilde{c}_j(Z_{max}, t) = \int_{0}^{2\pi} \int_{R-\tau}^{R} c_j(z, r, t) r dr d\theta \qquad \text{in } \Omega_{Crypt} $$

where $q_j$ is the absorption per enterocyte cell of the compound $j$ and $\tau$ defines the width of an annular region adjacent to the boundary $\Gamma_m$.
Remaining solutes and dissolved compounds are subject to homogeneous Robin boundary conditions, as no exchange occurs between the colon lumen and the crypt. These interactions are summarized schematically in Figure 3, while model parameters and the computation of mucosal transport coefficients are detailed in Appendix Table SM2 and Appendix SM5, respectively.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Mucosa exchange. Schematic representation of the exchange mechanisms between the colon and the crypt models.</figcaption></figure>

The coupling framework also accounts for biological signaling via Pattern Recognition Receptors (PRRs), which modulate stem and progenitor cell division to mediate host-microbiota symbiosis (10). In this integrated model, cell division is positively stimulated by the bacterial volume fraction $f_B$ near the mucosal surface. This coupling variable is calculated by integrating bacterial densities over an annular region adjacent to the mucosal wall:

$$ (2.27) \qquad f_B = \int_0^{2\pi} \int_{R-\tau}^R \sum_{i \in [bact]} f_i r \, dr \, d\theta $$

with $\tau$ defined as in Equation (2.26). The impact of this signal on crypt dynamics is detailed in Appendix SM6.

**2.5 Numerical Methods** The system of equations for the colon domain has been discretized and solved in Python using the FEniCSx finite element library (3), coupled with an in-house Python solver for the microscopic crypt model.

For the colon, the triangular mesh displayed in Figure 4 was specifically designed to face the computational challenges to accurately capture cross-scale flux exchanges while preserving numerical stability at the mucosal interface. The mesh refinement follows a three-level predefined resolution: a fine mesh manually specified near the mucus layer and a progressively coarser discretization toward the lumen. The crypt model uses a regular discretization with constant interval size (10).

The computational scheme adopted involves two successive executions for each time step (see Algorithm 2.1):

1. **Crypt Model:** Solved via a discrete kinetic scheme (1).

2. **Colon Model:** Solved through a multi-step sequence that includes solving:

* Stokes equations for fluid dynamics.

* ADR equations for $O_2$ and AMPs.

* ADR equations for all remaining entities.

$O_2$ and AMPs are solved independently because their crypt-derived concentrations are several orders of magnitude higher than other luminal solutes. This decoupling prevents ill-conditioning and ensures numerical convergence.
<figure><img src="image_5.png" /><figcaption>Fig. 4: **Computational mesh.** Full colon domain with zoomed view showing mesh refinement.</figcaption></figure>

While the ADR problems utilize a standard Krylov solver without preconditioning, the Stokes equations are solved using the MINRES method with an HYPRE BoomerAMG (8) algebraic multigrid preconditioner for accelerated convergence.

<caption>**Algorithm 2.1** Pseudo-code for solving the equations</caption>

**Input:** Maximum time $T_{\text{max}}$, Time step $\Delta t$
**Output:** Solutions for the crypt model and colon model
**while** $t < T_{\text{max}}$ **do**
  Solve the **crypt model**
  Solve the **colon model**
  Solve the **Stokes equation** for fluid flow in the colon
  Solve the **ADR equations** for AMPs and $O_2$
  Solve the **ADR equations** for all other entities
  Advance time: $t \leftarrow t + \Delta t$
**end while**

**3 Numerical Results** This section presents numerical results from the extended coupled colon–crypt model, parameterized to approximate the human proximal environment. A reference healthy diet (10) serves as the baseline for healthy gut dynamics, with all parameters detailed in Appendix SM1. Unless otherwise specified, results refer to steady-state solutions.

Validation is based on the three-average-crypt architecture introduced in Section 2. In addition, using this framework, we examine perturbed scenarios where the central crypt (Crypt 2) is impaired while the outer crypts (Crypt 1 and 3) remain healthy. Two pathological conditions are investigated: (i) elevated basal oxygen concentration (Section 3.3) and (ii) defective mucus production (Section 3.4).

*Units* The model scales space in centimeters (cm) and time in days (d). Since mixture components are defined as dimensionless volume fractions, reaction rates $F_i$ are expressed in $d^{-1}$. Dissolved compound densities are given in $\text{mol}\cdot\text{cm}^{-3}$. To
facilitate comparison with microbiological literature (CFU per mass unit), we assume an average bacterial mass of 5 pg and a density of 1 g·cm⁻³ (20). Detailed units for all parameters are provided in Appendix SM1.

**3.1 Physiological Reference-State Evaluation** To initially validate the model, we examine the cellular dynamics and structural organization of the intestinal crypt. The crypt requires 7 days to reach a steady-state population, a timescale consistent with the 4–5 day turnover rate of mammalian intestinal epithelium (21).

Figure 5a illustrates the spatial distribution of cell densities from the crypt base ($z = 0$) to the top ($z = Z_{max}$). By integrating this density profiles over the 1D domain, our model predicts a total of 2352 cells per crypt, aligning with literature estimates of 2427.8 ± 50.44 (4). Specifically, the model yields 635 proliferative cells (stem and progenitors) and 1681 differentiated cells, both within the reported ranges of 623.9 ± 234.1 and 1768.2 ± 434.5, respectively. The goblet-to-enterocyte ratio is approximately 1:3, consistent with physiological expectations. Furthermore, normalized solute concentrations along the crypt (Figure 5b) align with previously established results (10).

<figure><img src="image_6.png" /><figcaption>Fig. 5: (a) Cell densities along the crypt and (b) Normalized solutes concentration along the crypt. Cell density profiles are shown as a function of crypt height. The total number of cells for each cell type is the following: sc: 65, dcs: 36, pc: 570, ent: 1261, gc: 420.</figcaption></figure>

Following cellular validation, we evaluate the colon model's structural features, focusing on the spatial distribution of the newly introduced dual-layer mucus structure. We examine the volume fractions and cross-sections of soft and hard mucus to characterize their thickness and distinct physical behaviors. Due to the lack of human data (12), measurements from murine models were rescaled to fit the human physiological scale. The extended model is validated against the framework proposed in (15) and experimental observations.

Figure 6a illustrates the steady-state soft mucus distribution. The peak volume fraction ($f_{sm}^{max} \approx 7.3 \times 10^{-2}$) and layer thickness ($h_{sm} = 0.12$ cm, Figure SM2) closely match results from (15). The profile increases along the colon, consistent with *in vivo* observations (11). This confirms that the novel hard mucus layer and lateral mucus production mechanisms (Eqs. (2.20)–(2.21)) preserve established soft mucus dynamics.

The hard mucus layer (Figure 6a) exhibits a lower peak ($f_{hm}^{max} \approx 1.7 \times 10^{-2}$) and
a narrower thickness ($h_{hm} = 0.065$ cm, Figure SM3), in agreement with (11; 18). This behavior reflects its role as a localized, protective barrier.

These findings confirm that the coupled model effectively captures the distinct biological roles of both mucus layers.

**3.2 Bacterial Spatial Distribution Across Functional Groups** A key novelty of this coupled model is the explicit representation of the hard mucus layer and its associated radial repulsive force acting on solid components (Eq. (2.6)). This barrier mechanism confines bacteria to the lumen and soft mucus region, preventing epithelial contact while allowing the passage of liquid and dissolved compounds.

To analyze the overall system behavior, we consider the total bacterial population, $f_{bact}^{sum} = \sum_{i \in [bact]} f_i$. Given the difficulty of obtaining precise quantitative measurements in the colon, model validation focuses on the spatial distribution of bacterial entities, ensuring consistency with established biological observations.

<figure><img src="image_7.png" /><figcaption>Fig. 6: **Spatial distribution of soft mucus, hard mucus and total bacterial population within the colon domain (left panels); overlay of their normalized profiles (right panel).** Solid, dashed, and dotted lines represent total bacteria, soft, and hard mucus, respectively. Volume fractions are normalized by dividing them by their respective maximum values. The shaded blue and red regions highlight the spatial extent of the soft and hard mucus layers, respectively.</figcaption></figure>

Figure 6a illustrates the steady-state distribution of the total bacterial population and its radial sections. As expected, bacteria accumulate progressively downstream. A narrow dark-blue band at the epithelial boundary ($r \in [2.435, 2.50]$ cm) indicates the hard mucus layer, where bacteria are completely excluded due to the repulsive radial velocity term (Eq. (2.10)). Total density is nearly uniform in the lumen but increases steeply within the nutrient-rich soft mucus, reaching a maximum before dropping to zero at the hard mucus interface (Figure SM4). This is further visualized in Figure 6b, where the normalized bacterial radial section is overlaid with both mucus layers cross-sectional distributions normalized. The peaks within the soft mucus indicate bacterial accumulation where growth substrates are most available. The hard mucus layer acts as an effective barrier, preventing epithelial penetration.
Figure 6b also highlights the distinct structural behaviors of the two layers: the soft mucus decays gradually and extends toward the lumen, suggesting greater diffusive spread, whereas the hard mucus decays sharply and remains confined near the epithelium. This confinement is consistent with the attractive radial force (Eq. (2.11)) that anchors the hard mucus and limits its luminal dispersion.

As a final validation step, we analyze the velocity field computed *via* the Stokes equations (Figure 7). The flow exhibits a parabolic-like profile with maximum velocity along the centerline and a minimum at the walls, consistent with the no-slip condition (Eq. (2.18)). The average domain velocity, $\bar{u} \approx 85 \text{ cm} \cdot \text{d}^{-1}$, aligns with values reported in (15). Horizontal cross-sections confirm the velocity decay near the boundaries and highlight the influence of non-Newtonian fluid behavior and mucosal geometry on the local flow. Despite these spatial variations, the bacterial population remains permanently established, indicating that the flow conditions support stable microbial colonization without washout.

<figure><img src="image_8.png" /><figcaption>Fig. 7: Velocity distribution (left panel) and flow profiles (right panel). The color map on the left panel indicates the velocity magnitude. The superimposed vector field shows the flow direction, with arrow thickness proportional to the local velocity. The right panel compares the inlet velocity (dotted line) with the mid cross-section velocity profile (solid line), taken at the mid-height.</figcaption></figure>

**3.3 Scenario A: Oxygen Bloom** This scenario investigates the localized impact of an oxygen bloom resulting from increased extravasation at the base of the impaired crypt (Average Crypt 2). Specifically, the basal oxygen concentration was increased from $\tilde{c}_{O_2}(0, t) = 48.8$ to 150 [a.u.], following the approach in (10).

Steady-state oxygen profiles for the baseline and oxygen bloom scenarios are compared in Figure 8. Longitudinal sections at the mucosal wall ($r = R$) confirm the formation of an oxygen-rich microenvironment: the impaired crypt (Figure 8b) delivers a significantly higher oxygen flux into the colon compared to the healthy baseline (Figure 8a). Minor discontinuities in the profiles are numerical artifacts originating from the transitions between crypts in the domain architecture.

Spatial differences in oxygen distribution are further elucidated by the cross-sections in Figure 9. In colon segment 2 (Figure 9b), corresponding to the impaired crypt, $O_2$ concentrations are markedly elevated, consistent with the increased basal
<figure><img src="image_9.png" /><figcaption>Fig. 8: Steady-state oxygen concentration (left panels) and corresponding longitudinal sections at the mucosal wall ($r = R$) at the impaired crypt (right panels) for the healthy (a) and oxygen bloom (b) scenarios.</figcaption></figure>

input ($\tilde{c}_{O_2}(0, t) = 150$ vs 48.8 [a.u.]). Conversely, in segment 3 (Figure 9c), oxygen levels rapidly return to baseline, demonstrating the localized nature of the perturbation. This localized behavior is further confirmed by the concentrations measured at the top of the crypts ($z = Z_{max}$). In the average crypt corresponding to colon segment 1 (Figure SM5a), oxygen remains at the baseline $\tilde{c}_{O_2}(Z_{max}, t) \approx 5$ [a.u.]. Conversely, the average crypt related to segment 2 (Figure SM5b) exhibits a rapid increase, stabilizing at $\tilde{c}_{O_2}(Z_{max}, t) \approx 65$ [a.u.], while the measurement for the Average Crypt 3 (Figure SM5c) returns to values analogous to the healthy case. These measurements confirm the overall oxygen distribution observed within the colon lumen.

<figure><img src="image_10.png" /><figcaption>Fig. 9: Composite view of $O_2$ dynamics across the three colon segments. Each panel presents the oxygen concentration cross-sections inside the colon for the healthy (dotted line) and impaired (solid line) cases.</figcaption></figure>

The oxygen bloom directly perturbs the total anaerobic microbial populations. As illustrated in Figure 10, anaerobic abundance begins to decline in colon section 1 (Figure 10a), with a significantly more pronounced reduction in colon section 2
(Figure 10b) where the oxygen bloom is localized. In colon section 3 (Figure 10c), although the reduction remains evident, the abundance shows a trend toward recovery, converging toward the healthy baseline. This sensitivity confirms the vulnerability of these species to localized oxygen exposure. Moreover, while the perturbation is spatially centered in colon section 2, its impact extends to the adjacent regions.

Quantitatively, the total anaerobic population across the domain drops from 577 billion to 389 billion, with a significant reduction of 188 billion bacteria. Conversely, facultative bacteria remain resilient to localized oxygen variations, maintaining stable concentrations across all regions. This substantial decline in anaerobes, coupled with the resilience of facultative species, underscores how crypt-level oxygen enrichment propagates into the lumen, reshaping the spatial distribution of oxygen-sensitive communities.

<figure><img src="image_11.png" /><figcaption>Fig. 10: Spatial distribution of bacterial populations across the colon. Figures (a-c) show the cross-sectional abundance of Anaerobic Bacteria. In each figure, the impaired scenario (solid line) is compared against the healthy baseline case (dotted line).</figcaption></figure>

At the microscopic scale, Average Crypt 2 represents the most critical scenario due to the localized oxygen bloom. As shown in Figure SM7b, metabolite concentrations in the impaired crypt undergo significant alterations compared to both the healthy state (Figure 5b) and Average Crypt 1 (Figure SM6). While in healthy conditions oxygen is typically depleted toward the base ($z = 0$), the impaired crypt maintains high normalized $O_2$ levels throughout its domain, reaching approximately 0.4 at the base. This perturbation correlates with a shift in cell densities (Figure SM7a). We observe an expansion of progenitor (615 vs 569) and stem cell populations (85 vs 65), alongside a decrease in enterocytes (1225 vs 1261) and goblet cells (408 vs 420). This suggests that the oxygen-rich environment promotes an immature crypt state, where stem and progenitor cells fail to differentiate correctly during their upward migration. Conversely, in Average Crypt 3 (Figure SM8), oxygen levels drop and profiles converge back to healthy values (Figure 5), confirming the localized nature of the impairment.

### 3.4 Scenario B: Dysfunction in Mucus Production
This scenario investigates the localized impact of goblet cell dysfunction on the double mucus layer, specifically within Average Crypt 2. The impaired basal production rate $q_m$ is modeled as:

$$ (3.1) \qquad q_m^{\text{dysf}}(z) = q_m \left(1 - M e^{-\left(\frac{z-z_0}{\sigma}\right)^2}\right) $$

where $M = 1$ is the maximum relative reduction, $z_0 = 4.5$ marks the center of the impaired region, and $\sigma = 1$ controls the spatial spread. The specific spatial
reduction in mucus production, modeled by the Gaussian profile in Eq. (3.1), creates a disruption centered at the impaired crypt. The remaining crypts remain at healthy baseline conditions. The resulting production profile is shown in Figure 11c.

Figure 11 illustrates the volume fraction profiles of hard (Figure 11a) and soft mucus (Figure 11b). The Gaussian-driven reduction in $q_m$ creates a severe thinning and localized drop in concentration over the impaired crypt, representing a barrier breach at this specific site. In contrast, the surrounding crypts maintain mucus distributions and thicknesses comparable to the baseline, confirming the localized nature of the barrier failure.

<figure><img src="image_12.png" /><figcaption>Fig. 11: Spatial distribution of the hard (a) and soft (b) mucus layers under localized crypt dysfunction at steady-state, along with (c) the impaired mucus production profile. The mucus production rate profile in the impaired crypt (solid line) shows a localized reduction centered at $z_0 = 4.5$. The dashed line represents the healthy baseline mucus production rate $q_m$ for comparison.</figcaption></figure>

A quantitative comparison of the soft mucus layer is provided in Figure 12, where three cross-sectional profiles, each corresponding to an individual average crypt, compare the dysfunctional state (impaired) against the healthy baseline (control). The control regions in colon segment 1 (Figure 12a) and colon segment 3 (Figure 12c) exhibit profiles that overlap with the healthy baseline (red dotted line), confirming preserved barrier functionality in neighboring areas. Conversely, the impaired crypt in colon segment 2 (Figure 12b) shows a pronounced depletion in soft mucus concentration, particularly near the mucosal wall. This significant reduction confirms the localized breakdown of the mucus barrier due to the modeled goblet cell dysfunction.

An analogous depletion occurs in the hard mucus layer (Figure SM9), where the impaired crypt exhibits a significant loss of the protective barrier compared to the healthy baseline. As illustrated in Figure 11a, this reduction allows bacteria to penetrate toward the epithelium, triggering host defense mechanisms. Specifically,
<figure><img src="image_13.png" /><figcaption>Fig. 12: Cross-sectional profiles of soft mucus volume fraction across the three average crypts under the reference diet. In each figure, the impaired scenario (solid line) is compared against the healthy baseline case (dotted line).</figcaption></figure>

bacterial infiltration into the hard mucus layer prompts epithelial cells to produce *AMPs* (Eq. (2.23)). AMPs are crucial components of the innate immune system, contributing to bacterial control and enhancing epithelial integrity. The spatial distribution of AMPs production across the three colon segments highlights this dynamic. While AMPs maintain basal levels in colon segment 1 (Figure SM10a), the impaired scenario induces a marked increase in colon segment 2 (Figure SM10b), localized near the epithelial layer. This upregulation is sustained, though slightly attenuated, in colon segment 3 (Figure SM10c), demonstrating a targeted innate immune response to contain the microbial threat originating from the mucus dysfunction.

Having established the upregulation of AMPs (Figure SM10), we investigate their protective role against bacterial infiltration into the mucus layers. Figure 13 compares the total bacterial volume fraction near the epithelium in the dysfunctional crypt region against a control case where the AMPs response is artificially deactivated. When the AMP-mediated response is inactive (Figure 13a), bacteria penetrate the hard mucus and accumulate at the epithelial wall. The magnified view and the color bar confirm this invasion, showing a non-zero minimum bacterial density ($f_{bact}^{sum,min} \approx 1.1 \times 10^{-5}$) within the ordinarily sterile layer. Conversely, when the AMP response is active (Figure 13b), bacteria are excluded from the previously invaded region. This protective effect extends beyond the immediate impairment site, demonstrating that the defense mechanism extends spatially to contain the microbial threat.

However, Figure 13b reveals a bacteria-free niche surrounding the impaired crypt, representing a potential ecological vulnerability. Pathogens or AMP-resistant bacteria could exploit this cleared path to penetrate the mucosal wall, mirroring the invasion observed in the absence of an immune response (Figure 13a). This highlights an inherent trade-off in the immune strategy: while local clearance contains most microbes, it may inadvertently create a physical channel for highly resilient invaders.

To provide a quantitative overview, Figure 14 compares the total bacterial concentration profiles across all three colonic sections for the healthy baseline, the impaired case with AMP response, and the impaired case without it. While profiles in colon segment 1 (Figure 14a) remain comparable across all scenarios, colon segment 2 (Figure 14b) illustrates the impact of the immune response. In the presence of AMPs, the bacterial concentration drops to zero at the mucosal wall, confirming successful clearance, while the overall density is significantly reduced. This protective effect persists in the post-bloom region (Figure 14c), where bacterial concentrations remain
<figure><img src="image_14.png" /><figcaption>Fig. 13: **Total bacterial volume fraction near the impaired crypt.** (a) Distribution in the absence of AMPs response; (b) distribution with an active AMPs response.</figcaption></figure>

lower than in the de-activated AMP scenario, underscoring the sustained effectiveness
of the innate AMPs immune defense.

<figure><img src="image_15.png" /><figcaption>Fig. 14: Total bacterial concentration across the radial thickness of the colon for three scenarios: with AMPs response (solid line), without AMPs response (dash-dot-dot line), and the healthy reference case (dotted line).</figcaption></figure>

At the crypt level, the results indicate no significant variation in the distribution or behavior of cell populations compared to the healthy control. This suggests that, under the modeled conditions, impaired mucus production alone is insufficient to drive major alterations in epithelial dynamics. Similarly, the metabolic microenvironment within the crypt remains largely unaffected despite the structural compromise of the mucus layer.

**4 Conclusions** This study successfully introduced and validated a novel multiscale computational framework that integrates colonic microbial dynamics with a dual-layer mucus architecture and the host defense mechanisms of the epithelial
crypts. Building upon existing models, we enhanced physiological realism by incor-
porating active cellular migration within the crypts, explicitly modeling the barrier
properties of the hard mucus layer, and integrating the host's innate immune response
through localized AMPs production.

Numerical simulations provided key insights into host-microbiota interactions un-
der localized perturbations. By simulating pathological conditions, specifically an
oxygen bloom (Scenario A) and defective mucus production (Scenario B), the model
revealed how localized pathological conditions compromise barrier function, leading
to the disruption of oxygen-sensitive microbial communities and their subsequent in-
filtration into the inner mucus layer. The model clearly demonstrated the protective
efficacy of the AMPs-mediated immune response. The upregulation of AMPs, trig-
gered by bacterial penetration, successfully prevents deeper invasion, effectively clear-
ing bacteria at the mucosal wall and confirming the critical role of innate immunity in
maintaining epithelial integrity. By establishing a sterile niche at the mucosal bound-
ary, the immune response ensures that cell populations and metabolic profiles within
the crypt remain stable despite the structural compromise of the mucus layers. This
suggests that host defense mechanisms act as a crucial protective barrier, isolating
the epithelial tissue from localized luminal perturbations and preserving physiological
homeostasis.

This integrative framework lays the groundwork for advanced gut microbiota
modeling, with future research prioritizing enhanced realism, computational efficiency,
and clinical applicability. We plan to refine the model's geometry to better approxi-
mate the colon's intricate physical structure and, crucially, to integrate more realistic
representations of microbial metabolism. Recent work (13) has demonstrated the ef-
fectiveness of coupling genome-scale metabolic models with spatio-temporal transport
equations to study microbial competition. This approach allows for a more accurate
description of how species interact and compete for limited resources within complex
environments. By incorporating patient-specific clinical inputs, such as individual mi-
crobiota compositions, immune profiles, and dietary intake, this metabolic framework
will ultimately enable tailored treatment strategies and precision medicine applica-
tions in gastrointestinal health.

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# SUPPLEMENTARY MATERIALS: A MULTISCALE COUPLED PDE MODEL OF HOST-MICROBIOTA INTERACTION: APPLICATION TO COLONIC MUCOSAL BARRIER DYNAMICS *

ELEONORA PASTREMOLI†‡, BÉATRICE LAROCHE†‡, AND LORENZO SALA†‡

## SM1 Colon model parameters

<table><caption>Table SM1: Model unknowns</caption><thead><tr><th>Unknown</th><th>Unit</th><th>Description</th></tr></thead><tbody><tr><td colspan="3"><strong>Mixture components</strong></td></tr><tr><td>f<sub>sm</sub></td><td>[-]</td><td>Soft mucus volume fraction</td></tr><tr><td>f<sub>hm</sub></td><td>[-]</td><td>Hard mucus volume fraction</td></tr><tr><td>f<sub>BmonAna</sub></td><td>[-]</td><td>Volume fraction of bacteria metabolizing monosaccharides and highly sensitive to oxygen</td></tr><tr><td>f<sub>BmonTol</sub></td><td>[-]</td><td>Volume fraction of bacteria metabolizing monosaccharides and moderately sensitive to oxygen</td></tr><tr><td>f<sub>BlaAna</sub></td><td>[-]</td><td>Volume fraction of bacteria metabolizing lactate and highly sensitive to oxygen</td></tr><tr><td>f<sub>BlaTol</sub></td><td>[-]</td><td>Volume fraction of bacteria metabolizing lactate and moderately sensitive to oxygen</td></tr><tr><td>f<sub>BlaFac</sub></td><td>[-]</td><td>Volume fraction of bacteria metabolizing lactate and less sensitive to oxygen</td></tr><tr><td>f<sub>BH2S</sub></td><td>[-]</td><td>Volume fraction of bacteria metabolizing H<sub>2</sub>S</td></tr><tr><td>f<sub>BH2a</sub></td><td>[-]</td><td>Volume fraction of bacteria producing acetate from H<sub>2</sub></td></tr><tr><td>f<sub>BH2m</sub></td><td>[-]</td><td>Volume fraction of bacteria producing methane from H<sub>2</sub></td></tr><tr><td>f<sub>r</sub></td><td>[-]</td><td>Volume fraction of digestible residuals</td></tr><tr><td>f<sub>pol</sub></td><td>[-]</td><td>Volume fraction of polysaccharides</td></tr><tr><td>f<sub>prot</sub></td><td>[-]</td><td>Volume fraction of dietary proteins</td></tr><tr><td>f<sub>l</sub></td><td>[-]</td><td>Volume fraction of liquid</td></tr><tr><td colspan="3"><strong>Diffusing compounds</strong></td></tr><tr><td>c<sub>mon</sub></td><td>mol.cm<sup>-3</sup></td><td>Concentration of monosaccharides</td></tr><tr><td>c<sub>lac</sub></td><td>mol.cm<sup>-3</sup></td><td>Concentration of lactate</td></tr><tr><td>c<sub>ac</sub></td><td>mol.cm<sup>-3</sup></td><td>Concentration of acetate</td></tr></tbody></table>

<footnote>†Université Paris-Saclay, INRAE, MaIAGE, 78350 Jouy-en-Josas, France.</footnote>

<footnote>‡Université Paris-Saclay, INRIA, MUSCA, 91120 Palaiseau, France.</footnote>

<footnote>*Submitted to the editors February 23, 2026.</footnote>
<table><caption>Table SM1 – Continued</caption><thead><tr><th>Unknown</th><th>Unit</th><th>Description</th></tr></thead><tbody><tr><td>c<sub>pro</sub></td><td>mol.cm<sup>-3</sup></td><td>Concentration of propionate</td></tr><tr><td>c<sub>but</sub></td><td>mol.cm<sup>-3</sup></td><td>Concentration of butyrate</td></tr><tr><td>c<sub>CH4</sub></td><td>mol.cm<sup>-3</sup></td><td>Concentration of methane</td></tr><tr><td>c<sub>CO2</sub></td><td>mol.cm<sup>-3</sup></td><td>Concentration of dissolved carbon dioxide</td></tr><tr><td>c<sub>H2</sub></td><td>mol.cm<sup>-3</sup></td><td>Concentration of dissolved hydrogen</td></tr><tr><td>c<sub>H2S</sub></td><td>mol.cm<sup>-3</sup></td><td>Concentration of hydrogen sulfide</td></tr><tr><td>c<sub>O2</sub></td><td>[a.u.]</td><td>Concentration of oxygen</td></tr><tr><td>c<sub>AMPs</sub></td><td>mol.cm<sup>-3</sup></td><td>Concentration of AMPs</td></tr><tr><td colspan="3"><strong>Source functions</strong></td></tr><tr><td>F<sub>i</sub></td><td>d<sup>-1</sup></td><td>Source function for the component i &isin; I<sub>S</sub></td></tr><tr><td>G<sub>j</sub></td><td>d<sup>-1</sup></td><td>Source function for the component j &isin; I<sub>D</sub></td></tr><tr><td colspan="3"><strong>Boundary conditions</strong></td></tr><tr><td>&gamma;<sub>f<sub>i</sub></sub></td><td>cm.d<sup>-1</sup></td><td>Boundary flux for component i &isin; I<sub>S</sub></td></tr><tr><td>&gamma;<sub>c<sub>j</sub></sub></td><td>mol.cm<sup>-2</sup>.d<sup>-1</sup></td><td>Boundary flux for component j &isin; I<sub>D</sub></td></tr></tbody></table>

<table><caption>Table SM2: Colon model parameters</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th><th>Refs</th></tr></thead><tbody><tr><td colspan="5"><strong>Colon geometry</strong></td></tr><tr><td>L</td><td>10</td><td>cm</td><td>Simulated colon length</td><td>-</td></tr><tr><td>R</td><td>2.5</td><td>cm</td><td>Proximal colon radius</td><td>(6)</td></tr><tr><td colspan="5"><strong>Diffusion parameters</strong></td></tr><tr><td>&sigma;<sub>mon</sub></td><td>1.4</td><td>cm.d<sup>-1</sup></td><td>Monosaccharides diffusion coefficient</td><td>(5)</td></tr><tr><td>&sigma;<sub>lac</sub></td><td>1.4</td><td>cm.d<sup>-1</sup></td><td>Lactate diffusion coefficient</td><td>(5)</td></tr><tr><td>&sigma;<sub>ac</sub></td><td>1.4</td><td>cm.d<sup>-1</sup></td><td>Acetate diffusion coefficient</td><td>(5)</td></tr><tr><td>&sigma;<sub>pro</sub></td><td>1.4</td><td>cm.d<sup>-1</sup></td><td>Propionate diffusion coefficient</td><td>(5)</td></tr><tr><td>&sigma;<sub>but</sub></td><td>1.4</td><td>cm.d<sup>-1</sup></td><td>Butyrate diffusion coefficient</td><td>(5)</td></tr><tr><td>&sigma;<sub>CH4</sub></td><td>2</td><td>cm.d<sup>-1</sup></td><td>CH<sub>4</sub> diffusion coefficient</td><td>(5)</td></tr><tr><td>&sigma;<sub>CO2</sub></td><td>2</td><td>cm.d<sup>-1</sup></td><td>CO<sub>2</sub> diffusion coefficient</td><td>(5)</td></tr></tbody></table>
<table><caption>Table SM2 – Continued</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th><th>Refs</th></tr></thead><tbody><tr><td>σ<sub>H2</sub></td><td>0.8</td><td>cm.d<sup>-1</sup></td><td>H<sub>2</sub> diffusion coefficient</td><td>(5)</td></tr><tr><td>σ<sub>H2S</sub></td><td>0.8</td><td>cm.d<sup>-1</sup></td><td>H<sub>2</sub>S diffusion coefficient</td><td>-</td></tr><tr><td>σ<sub>O2</sub></td><td>0.036 × 24</td><td>cm.d<sup>-1</sup></td><td>O<sub>2</sub> diffusion coefficient</td><td>(2)</td></tr><tr><td>σ<sub>AMPs</sub></td><td>5 × 10<sup>-3</sup> × 24</td><td>cm.d<sup>-1</sup></td><td>AMPs diffusion coefficient</td><td>(3)</td></tr><tr><td>σ<sub>hm</sub></td><td>0.8</td><td>cm.d<sup>-1</sup></td><td>Hard mucus diffusion coefficient</td><td>-</td></tr><tr><td>σ</td><td>0.8</td><td>cm.d<sup>-1</sup></td><td>Other solid components diffusion coefficient</td><td>(5)</td></tr><tr><td colspan="5"><strong>Viscosity parameters</strong></td></tr><tr><td>μ<sub>l,min</sub></td><td>0.864 × 10<sup>3</sup></td><td>g.cm<sup>-1</sup>.d<sup>-1</sup></td><td>Water viscosity</td><td>(1)</td></tr><tr><td>μ<sub>l,max</sub></td><td>36.7 × 10<sup>3</sup></td><td>g.cm<sup>-1</sup>.d<sup>-1</sup></td><td>Viscosity of dried mixture</td><td>(5)</td></tr><tr><td>f<sub>l,thr</sub></td><td>0.52</td><td>[-]</td><td>Viscosity threshold: liquid sigmoid</td><td>(5)</td></tr><tr><td>α<sub>l</sub></td><td>4</td><td>[-]</td><td>Stiffness: liquid sigmoid</td><td>(5)</td></tr><tr><td>μ<sub>m,min</sub></td><td>0.864 × 10<sup>3</sup></td><td>g.cm<sup>-1</sup>.d<sup>-1</sup></td><td>Minimal mucus viscosity</td><td>(1)</td></tr><tr><td>μ<sub>m,max</sub></td><td>73.4 × 10<sup>3</sup></td><td>g.cm<sup>-1</sup>.d<sup>-1</sup></td><td>Maximal mucus viscosity</td><td>(1)</td></tr><tr><td>f<sub>m,thr</sub></td><td>0.0085</td><td>[-]</td><td>Viscosity threshold: mucus sigmoid</td><td>-</td></tr><tr><td>α<sub>m</sub></td><td>14</td><td>[-]</td><td>Stiffness: mucus sigmoid</td><td>(5)</td></tr><tr><td colspan="5"><strong>Force Function parameters</strong></td></tr><tr><td>A<sub>top</sub></td><td>20</td><td>[-]</td><td>Maximum value for attractive force</td><td>-</td></tr><tr><td>A<sub>bot</sub></td><td>5</td><td>[-]</td><td>Minimum value for attractive force</td><td>-</td></tr><tr><td>C<sub>top</sub></td><td>55</td><td>[-]</td><td>Maximum value for repulsive force</td><td>-</td></tr><tr><td>C<sub>bot</sub></td><td>0</td><td>[-]</td><td>Maximum value for repulsive force</td><td>-</td></tr><tr><td>thr</td><td>0.01</td><td>[-]</td><td>Threshold for hard mucus to form a barrier</td><td>-</td></tr><tr><td>k</td><td>1000</td><td>[-]</td><td>Slope coefficient</td><td>-</td></tr><tr><td colspan="5"><strong>Input on Γ<sup>in</sup></strong></td></tr><tr><td>u<sub>in</sub></td><td></td><td>cm.d<sup>-1</sup></td><td>Half average velocity inflow</td><td>-</td></tr><tr><td>f<sup>in</sup><sub>sm</sub></td><td>f<sub>sm</sub>(0, r, z)</td><td>[-]</td><td>Soft mucus density input</td><td>-</td></tr></tbody></table>
<table><caption>Table SM2 – Continued</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th><th>Refs</th></tr></thead><tbody><tr><td>f<sup>in</sup><sub>pol</sub></td><td>0.05</td><td>[-]</td><td>Polysaccharide density input</td><td>(3)</td></tr><tr><td>f<sup>in</sup><sub>prot</sub></td><td>1.35 × 10<sup>-2</sup></td><td>[-]</td><td>Protein density input</td><td>(3)</td></tr><tr><td>f<sup>in</sup><sub>r</sub></td><td>0.05</td><td>[-]</td><td>Dietary residual density input</td><td>(3)</td></tr><tr><td>f<sup>in</sup><sub>B<sup>φ</sup><sub>mon</sub></sub></td><td>6.1 × 10<sup>-5</sup> × 1/2</td><td>[-]</td><td>B<sup>φ</sup><sub>mon</sub> density input</td><td>(6)</td></tr><tr><td>f<sup>in</sup><sub>B<sup>η</sup><sub>mon</sub></sub></td><td>6.1 × 10<sup>-5</sup> × 1/2</td><td>[-]</td><td>B<sup>η</sup><sub>mon</sub> density input</td><td>(6)</td></tr><tr><td>f<sup>in</sup><sub>B<sup>φ</sup><sub>la</sub></sub></td><td>2.03 × 10<sup>-5</sup> × 1/3</td><td>[-]</td><td>B<sup>φ</sup><sub>la</sub> density input</td><td>(6)</td></tr><tr><td>f<sup>in</sup><sub>B<sup>η</sup><sub>la</sub></sub></td><td>2.03 × 10<sup>-5</sup> × 1/3</td><td>[-]</td><td>B<sup>η</sup><sub>la</sub> density input</td><td>(6)</td></tr><tr><td>f<sup>in</sup><sub>B<sup>δ</sup><sub>la</sub></sub></td><td>2.03 × 10<sup>-5</sup> × 1/3</td><td>[-]</td><td>B<sup>δ</sup><sub>la</sub> density input</td><td>(6)</td></tr><tr><td>f<sup>in</sup><sub>B<sup>δ</sup><sub>H2S</sub></sub></td><td>1.01 × 10<sup>-5</sup></td><td>[-]</td><td>B<sup>δ</sup><sub>H2S</sub> density input</td><td>(3)</td></tr><tr><td>f<sup>in</sup><sub>B<sup>φ</sup><sub>H2a</sub></sub></td><td>1.01 × 10<sup>-5</sup></td><td>[-]</td><td>B<sup>φ</sup><sub>H2a</sub> density input</td><td>(6)</td></tr><tr><td>f<sup>in</sup><sub>B<sup>φ</sup><sub>H2m</sub></sub></td><td>1.01 × 10<sup>-5</sup></td><td>[-]</td><td>B<sup>φ</sup><sub>H2m</sub> density input</td><td>(6)</td></tr><tr><td>c<sup>in</sup><sub>mon</sub></td><td>3.33 × 10<sup>-5</sup></td><td>mol.cm<sup>-3</sup></td><td>Monosaccharide density input</td><td>(6)</td></tr><tr><td colspan="5"><strong>Flux on Γ<sup>m</sup></strong></td></tr><tr><td>q<sub>m</sub></td><td>1.2 × 10<sup>-8</sup></td><td>cm.d<sup>-1</sup>.cell<sup>-1</sup></td><td>Proximal mucus production rate per day per cell</td><td>(6)</td></tr><tr><td>α<sub>sm</sub></td><td>0.952</td><td>[-]</td><td>Proportion of soft mucus produced</td><td>-</td></tr><tr><td>α<sub>hm</sub></td><td>0.048</td><td>[-]</td><td>Proportion of hard mucus produced</td><td>-</td></tr><tr><td>q<sub>l</sub></td><td>1.91 × 10<sup>-10</sup></td><td>cm<sup>-2</sup>.d<sup>-1</sup>.cell<sup>-1</sup></td><td>Proximal liquid absorption rate per day per cell</td><td>(6)</td></tr><tr><td>q<sub>H2S</sub></td><td>4.95 × 10<sup>-9</sup></td><td>cm<sup>-2</sup>.d<sup>-1</sup>.cell<sup>-1</sup></td><td>H<sub>2</sub>S absorption rate per day per cell</td><td>(3)</td></tr><tr><td>q<sub>la</sub></td><td>1.61 × 10<sup>-9</sup></td><td>cm<sup>-2</sup>.d<sup>-1</sup>.cell<sup>-1</sup></td><td>Lactate absorption rate per day per cell</td><td>(6)</td></tr><tr><td>q<sub>ac</sub></td><td>2.41 × 10<sup>-9</sup></td><td>cm<sup>-2</sup>.d<sup>-1</sup>.cell<sup>-1</sup></td><td>Acetate absorption rate per day per cell</td><td>(6)</td></tr><tr><td>q<sub>pro</sub></td><td>1.96 × 10<sup>-9</sup></td><td>cm<sup>-2</sup>.d<sup>-1</sup>.cell<sup>-1</sup></td><td>Propionate absorption rate per day per cell</td><td>(6)</td></tr><tr><td>q<sub>but</sub></td><td>1.64 × 10<sup>-9</sup></td><td>cm<sup>-2</sup>.d<sup>-1</sup>.cell<sup>-1</sup></td><td>Butyrate absorption rate per day per cell</td><td>(6)</td></tr><tr><td>q<sub>AMPs</sub></td><td>1.96 × 10<sup>-18</sup></td><td>cm.d<sup>-1</sup>.cell<sup>-1</sup></td><td>AMPs basal production rate per day per cell</td><td>(3)</td></tr></tbody></table>
<table><caption>Table SM2 – Continued</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th><th>Refs</th></tr></thead><tbody><tr><td>q<sup>imm</sup><sub>AMPs</sub></td><td>5 × 10<sup>-7</sup></td><td>cm<sup>3</sup>·d<sup>-1</sup>·cell<sup>-1</sup></td><td>AMPs production rate per cell induced by immune response</td><td>-</td></tr><tr><td>C<sub>bact</sub></td><td>10<sup>-4</sup></td><td>[-]</td><td>Bacterial threshold to trigger the immune AMPs response</td><td>-</td></tr><tr><td>d</td><td>0.025</td><td>cm</td><td>Annular width within the hard mucus layer</td><td>-</td></tr><tr><td>g<sub>O2</sub></td><td>0.036 × 24</td><td>cm·d<sup>-1</sup></td><td>O<sub>2</sub> diffusion rate per day</td><td>(6)</td></tr><tr><td>τ</td><td>0.1</td><td>cm</td><td>Annular width</td><td>-</td></tr><tr><td>N<sub>crypt</sub></td><td>200 × 100</td><td>d<sup>-1</sup></td><td>Crypt density per surface unit</td><td>(2)</td></tr><tr><td>N<sub>gc</sub></td><td>420</td><td>[-]</td><td>Estimated number of Goblet cells per healthy crypt</td><td>(3)</td></tr><tr><td>N<sub>ent</sub></td><td>1270</td><td>[-]</td><td>Estimated number of Enterocytes per healthy crypt</td><td>(3)</td></tr><tr><td>U<sub>per,r</sub></td><td>-0.1</td><td>cm·d<sup>-1</sup></td><td>Radial peristaltic velocity</td><td>(5)</td></tr><tr><td>U<sub>per,z</sub></td><td>-10</td><td>cm·d<sup>-1</sup></td><td>Longitudinal peristaltic velocity</td><td>(5)</td></tr></tbody></table>

<table><caption>Table SM3: Source function</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th><th>Refs</th></tr></thead><tbody><tr><td colspan="5" style="text-align: center; font-weight: bold;">Maximum specific reaction rates</td></tr><tr><td>k<sup>P1</sup></td><td>1.20 × 10<sup>3</sup></td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P1</td><td>(6)</td></tr><tr><td>k<sup>P2sm</sup></td><td>70</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P2<sub>sm</sub></td><td>(3)</td></tr><tr><td>k<sup>P2prot</sup></td><td>90</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P2<sub>prot</sub></td><td>(3)</td></tr><tr><td>k<sup>P3</sup></td><td>7.92</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P3</td><td>(6)</td></tr><tr><td>k<sup>P4</sup></td><td>103</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P4</td><td>(6)</td></tr></tbody></table>
<table><caption>Table SM3 – Continued</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th><th>Refs</th></tr></thead><tbody><tr><td>k<sup>P5</sup></td><td>108.84</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P5</td><td>(6)</td></tr><tr><td>k<sup>P6</sup></td><td>22.58</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P6</td><td>(6)</td></tr><tr><td>k<sup>P7</sup></td><td>0.05 × 0.002</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P7</td><td>(3)</td></tr><tr><td>k<sup>P8,φ</sup></td><td>50</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P8</td><td>-</td></tr><tr><td>k<sup>P8,η</sup></td><td>10</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P8</td><td>-</td></tr><tr><td>k<sup>P9</sup></td><td>200</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P9</td><td>-</td></tr><tr><td>k<sup>P10</sup></td><td>0.01</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P10</td><td>(6)</td></tr><tr><td>k<sup>P11</sup></td><td>200</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P11</td><td>(6)</td></tr><tr><td>k<sup>P12</sup></td><td>1</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P12</td><td>-</td></tr><tr><td>k<sup>P13</sup></td><td>20</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P13</td><td>-</td></tr><tr><td>k<sup>P14</sup></td><td>200</td><td>d<sup>-1</sup></td><td>Max. specific reaction rate for P14</td><td>-</td></tr><tr><td colspan="5" style="text-align: center; font-weight: bold;">Half saturation constant</td></tr><tr><td>K<sub>f</sub><sup>P1</sup></td><td>29.99 / 113</td><td>[-]</td><td>Comtois law</td><td>(6)</td></tr><tr><td>K<sub>f</sub><sup>P2</sup></td><td>50/113</td><td>[-]</td><td>Comtois law</td><td>(3)</td></tr><tr><td>K<sub>s</sub><sup>P3</sup></td><td>2.6 × 10<sup>-6</sup></td><td>mol.cm<sup>-3</sup></td><td>Monod law</td><td>(6)</td></tr><tr><td>K<sub>s</sub><sup>P4</sup></td><td>6.626 × 10<sup>-6</sup></td><td>mol.cm<sup>-3</sup></td><td>Monod law</td><td>(6)</td></tr><tr><td>K<sub>s</sub><sup>P5</sup></td><td>1.7 × 10<sup>-6</sup></td><td>mol.cm<sup>-3</sup></td><td>Monod law</td><td>(6)</td></tr><tr><td>K<sub>s</sub><sup>P6</sup></td><td>1.563 × 10<sup>-6</sup></td><td>mol.cm<sup>-3</sup></td><td>Monod law</td><td>(6)</td></tr><tr><td colspan="5" style="text-align: center; font-weight: bold;">Toxicity parameters</td></tr><tr><td>K<sup>P7</sup></td><td>0.7 × 10<sup>-6</sup></td><td>mol.cm<sup>-3</sup></td><td>Activ. parameter for P7</td><td>(3)</td></tr><tr><td>κ<sup>P7</sup></td><td>0.6 × 10<sup>-6</sup></td><td>mol.cm<sup>-3</sup></td><td>Activ. parameter for P7</td><td>(3)</td></tr><tr><td>K<sup>P8,φ</sup></td><td>3.5</td><td>[a.u]</td><td>Activ. parameter for P8</td><td>(3)</td></tr><tr><td>κ<sup>P8,φ</sup></td><td>0.5</td><td>[a.u]</td><td>Activ. parameter for P8</td><td>(3)</td></tr><tr><td>K<sup>P8,η</sup></td><td>4</td><td>[a.u]</td><td>Activ. parameter for P8</td><td>(3)</td></tr><tr><td>κ<sup>P8,η</sup></td><td>1</td><td>[a.u]</td><td>Activ. parameter for P8</td><td>(3)</td></tr><tr><td>K<sup>P9</sup></td><td>4 × 10<sup>-8</sup></td><td>mol.cm<sup>-3</sup></td><td>Activ. parameter for P9</td><td>(3)</td></tr></tbody></table>
<table><caption>Table SM3 – Continued</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th><th>Refs</th></tr></thead><tbody><tr><td>κ<sup>P9</sup></td><td>2.5 × 10<sup>-8</sup></td><td>mol.cm<sup>-3</sup></td><td>Activ. parameter for P9</td><td>(3)</td></tr><tr><td colspan="5"><strong>pH influence</strong></td></tr><tr><td>I<sup>prox</sup><sub>pH</sub></td><td>0.3</td><td>[-]</td><td>pH influence for P6</td><td>(5)</td></tr><tr><td colspan="5"><strong>Focus on process 11</strong></td></tr><tr><td>K<sup>P11</sup><sub>H2S</sub></td><td>7.29 × 10<sup>-4</sup></td><td>mol.bar<sup>-1</sup></td><td>Henry's law coef.</td><td>(3)</td></tr><tr><td>K<sup>P11</sup><sub>CH4</sub></td><td>0.0011</td><td>mol.bar<sup>-1</sup></td><td>Henry's law coef.</td><td>(6)</td></tr><tr><td>K<sup>P11</sup><sub>CO2</sub></td><td>0.0255</td><td>mol.bar<sup>-1</sup></td><td>Henry's law coef.</td><td>(6)</td></tr><tr><td>K<sup>P11</sup><sub>H2</sub></td><td>7.29 × 10<sup>-4</sup></td><td>mol.bar<sup>-1</sup></td><td>Henry's law coef.</td><td>(6)</td></tr><tr><td>R</td><td>0.08314</td><td>bar.mol<sup>-1</sup>.K<sup>-1</sup></td><td>Ideal gas constant</td><td>(6)</td></tr><tr><td>T</td><td>310.15</td><td>K</td><td>Gas absolute temperature</td><td>(6)</td></tr><tr><td>[H<sub>2</sub>S<sub>g</sub>]<sub>∞</sub></td><td>3.6505 × 10<sup>-7</sup></td><td>mol.cm<sup>-3</sup></td><td>Gaseous CH<sub>4</sub> steady state level</td><td>(3)</td></tr><tr><td>[CH<sub>4,g</sub>]<sub>∞</sub></td><td>1.9106 × 10<sup>-10</sup></td><td>mol.cm<sup>-3</sup></td><td>Gaseous CH<sub>4</sub> steady state level</td><td>(6)</td></tr><tr><td>[CO<sub>2,g</sub>]<sub>∞</sub></td><td>1.19 × 10<sup>-5</sup></td><td>mol.cm<sup>-3</sup></td><td>Gaseous CO<sub>2</sub> steady state level</td><td>(6)</td></tr><tr><td>[H<sub>2,g</sub>]<sub>∞</sub></td><td>3.6505 × 10<sup>-7</sup></td><td>mol.cm<sup>-3</sup></td><td>Gaseous H<sub>2</sub> steady state level</td><td>(6)</td></tr><tr><td colspan="5"><strong>Process P1</strong></td></tr><tr><td>Y<sup>P1</sup><sub>sm</sub></td><td>8.850 × 10<sup>-3</sup></td><td>[-]</td><td>Yield of sm in process 1</td><td>(6)</td></tr><tr><td>Y<sup>P1</sup><sub>pol</sub></td><td>8.850 × 10<sup>-3</sup></td><td>[-]</td><td>Yield of pol in process 1</td><td>(6)</td></tr><tr><td>Y<sup>P1</sup><sub>mon</sub></td><td>4.425 × 10<sup>-5</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of mon in process 1</td><td>(6)</td></tr><tr><td colspan="5"><strong>Process P2</strong></td></tr><tr><td>Y<sup>P2</sup><sub>sm</sub></td><td>8.850 × 10<sup>-3</sup></td><td>[-]</td><td>Yield of sm in process 2</td><td>-</td></tr><tr><td>Y<sup>P2</sup><sub>prot</sub></td><td>8.850 × 10<sup>-3</sup></td><td>[-]</td><td>Yield of prot in process 2</td><td>(3)</td></tr><tr><td>Y<sup>P2sm</sup><sub>B<sup>δ</sup><sub>H2S</sub></sub></td><td>0.001</td><td>mol.cm<sup>-3</sup></td><td>Production yield of B<sup>δ</sup><sub>H2S</sub> in process 2 specific to mu-coproteins</td><td>(3)</td></tr><tr><td>Y<sup>P2prot</sup><sub>B<sup>δ</sup><sub>H2S</sub></sub></td><td>0.01</td><td>mol.cm<sup>-3</sup></td><td>Production yield of B<sup>δ</sup><sub>H2S</sub> in process 2 specific to proteins</td><td>(3)</td></tr><tr><td>Y<sup>P2sm</sup><sub>H2S</sub></td><td>7.08 × 10<sup>-5</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of H<sub>2</sub>S in process 2 specific to mu-coproteins</td><td>(3)</td></tr></tbody></table>
<table><caption>Table SM3 – Continued</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th><th>Refs</th></tr></thead><tbody><tr><td>Y<sup>P2<sub>prot</sub></sup><sub>H<sub>2</sub>S</sub></td><td>1.33 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of H<sub>2</sub>S in process 2 specific to proteins</td><td>(3)</td></tr><tr><td>Y<sup>P2</sup><sub>H<sub>2</sub></sub></td><td>-2.83 × 10<sup>-4</sup></td><td>mol.cm<sup>-3</sup></td><td>Yield of H<sub>2</sub> in process 2</td><td>(3)</td></tr><tr><td>Y<sup>P2</sup><sub>O<sub>2</sub></sub></td><td>-200</td><td>[a.u]</td><td>Yield of O<sub>2</sub> in process 2</td><td>(3)</td></tr><tr><td>Y<sup>P2</sup><sub>CO<sub>2</sub></sub></td><td>-7.08 × 10<sup>-5</sup></td><td>mol.cm<sup>-3</sup></td><td>Yield of CO<sub>2</sub> in process 2</td><td>(3)</td></tr><tr><td colspan="5"><strong>Process P3</strong></td></tr><tr><td>Y<sup>P3</sup><sub>mon</sub></td><td>8.850 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Degradation yield of mon in process 3</td><td>(6)</td></tr><tr><td>Y<sup>P3</sup><sub>B<sub>mon</sub></sub></td><td>0.120</td><td>[-]</td><td>Biomass yield factor for B<sub>mon</sub></td><td>(6)</td></tr><tr><td>Y<sup>P3</sup><sub>la</sub></td><td>4.416 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of la in process 3</td><td>(6)</td></tr><tr><td>Y<sup>P3</sup><sub>ac</sub></td><td>5.18 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of ac in process 3</td><td>(6)</td></tr><tr><td>Y<sup>P3</sup><sub>pro</sub></td><td>2.124 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of pro in process 3</td><td>(6)</td></tr><tr><td>Y<sup>P3</sup><sub>but</sub></td><td>2.389 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of but in process 3</td><td>(6)</td></tr><tr><td>Y<sup>P3</sup><sub>CO<sub>2</sub></sub></td><td>9.735 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of CO<sub>2</sub> in process 3</td><td>(6)</td></tr><tr><td>Y<sup>P3</sup><sub>H<sub>2</sub></sub></td><td>1.274 × 10<sup>-2</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of H<sub>2</sub> in process 3</td><td>(6)</td></tr><tr><td colspan="5"><strong>Process P4</strong></td></tr><tr><td>Y<sup>P4</sup><sub>la</sub></td><td>8.850 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Degradation yield of la in process 4</td><td>(6)</td></tr><tr><td>Y<sup>P4</sup><sub>B<sub>la</sub></sub></td><td>0.120</td><td>[-]</td><td>Biomass yield factor for B<sub>la</sub></td><td>(6)</td></tr><tr><td>Y<sup>P4</sup><sub>ac</sub></td><td>1.177 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of ac in process 4</td><td>(6)</td></tr><tr><td>Y<sup>P4</sup><sub>pro</sub></td><td>2.363 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of pro in process 4</td><td>(6)</td></tr><tr><td>Y<sup>P4</sup><sub>but</sub></td><td>1.770 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of but in process 4</td><td>(6)</td></tr><tr><td>Y<sup>P4</sup><sub>CO<sub>2</sub></sub></td><td>4.717 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of CO<sub>2</sub> in process 4</td><td>(6)</td></tr><tr><td>Y<sup>P4</sup><sub>H<sub>2</sub></sub></td><td>3.540 × 10<sup>-2</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of H<sub>2</sub> in process 4</td><td>(6)</td></tr></tbody></table>
<table><caption>Table SM3 – Continued</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th><th>Refs</th></tr></thead><tbody><tr><td>Y<sub>O2</sub><sup>P4</sup></td><td>-300</td><td>[a.u]</td><td>Production yield of but in process 4</td><td>(6)</td></tr><tr><td colspan="5"><strong>Process P5</strong></td></tr><tr><td>Y<sub>H2</sub><sup>P5</sup></td><td>8.850 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Degradation yield of H<sub>2</sub> in process 5</td><td>(6)</td></tr><tr><td>Y<sub>B<sub>H2a</sub><sup>φ</sup></sub><sup>P5</sup></td><td>0.043</td><td>[-]</td><td>Biomass yield factor for B<sub>H2a</sub><sup>φ</sup></td><td>(6)</td></tr><tr><td>Y<sub>ac</sub><sup>P5</sup></td><td>2.21 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of ac in process 5</td><td>(7)</td></tr><tr><td>Y<sub>CO2</sub><sup>P5</sup></td><td>-4.424 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Yield of CO<sub>2</sub> in process 5</td><td>(6)</td></tr><tr><td colspan="5"><strong>Process P6</strong></td></tr><tr><td>Y<sub>H2</sub><sup>P6</sup></td><td>8.850 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Degradation yield of H<sub>2</sub> in process 6</td><td>(6)</td></tr><tr><td>Y<sub>B<sub>H2m</sub><sup>φ</sup></sub><sup>P6</sup></td><td>0.062</td><td>[-]</td><td>Biomass yield factor for B<sub>H2m</sub><sup>φ</sup></td><td>(6)</td></tr><tr><td>Y<sub>CH4</sub><sup>P6</sup></td><td>8.407 × 10<sup>-4</sup></td><td>mol.cm<sup>-3</sup></td><td>Production yield of CH<sub>4</sub> in process 6</td><td>(6)</td></tr><tr><td>Y<sub>CO2</sub><sup>P6</sup></td><td>-3.982 × 10<sup>-3</sup></td><td>mol.cm<sup>-3</sup></td><td>Yield of CO<sub>2</sub> in process 6</td><td>(6)</td></tr><tr><td colspan="5"><strong>Process P7</strong></td></tr><tr><td>Y<sub>sm</sub><sup>P7</sup></td><td>8.850 × 10<sup>-3</sup></td><td>[-]</td><td>Yield of sm in process 7</td><td>(6)</td></tr><tr><td colspan="5"><strong>Process P12</strong></td></tr><tr><td>Y<sub>hm</sub><sup>P12</sup></td><td>0.8</td><td>[-]</td><td>Yield of hm in process 12</td><td>-</td></tr><tr><td>Y<sub>sm</sub><sup>P12</sup></td><td>0</td><td>[-]</td><td>Yield of sm in process 12</td><td>-</td></tr><tr><td colspan="5"><strong>Process P13</strong></td></tr><tr><td>Y<sub>O2</sub><sup>P13</sup></td><td>1</td><td>[-]</td><td>Yield of O<sub>2</sub> in process 13</td><td>-</td></tr><tr><td colspan="5"><strong>Process P14</strong></td></tr><tr><td>Y<sub>AMPs</sub><sup>P14</sup></td><td>1</td><td>[-]</td><td>Yield of AMPs in process 14</td><td>-</td></tr></tbody></table>

<table><caption>Table SM4: Crypt model parameters</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th><th>Refs</th></tr></thead><tbody><tr><td colspan="5"><strong>Crypt geometry</strong></td></tr><tr><td>Z<sub>max</sub></td><td>400</td><td>μm</td><td>Height</td><td>(4)</td></tr></tbody></table>
<table><caption>Table SM4 – Continued</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th><th>Refs</th></tr></thead><tbody><tr><td>r<sub>0</sub></td><td>37</td><td>μm</td><td>Radius</td><td>(4)</td></tr><tr><td>ε</td><td>4</td><td>μm</td><td>Shape parameter</td><td>(3)</td></tr><tr><td colspan="5"><strong>Metabolites β-oxidation parameters</strong></td></tr><tr><td>s<sub>o,la</sub></td><td>3</td><td>-</td><td>O<sub>2</sub> stoichiometric coefficient during lactate consumption</td><td>(3)</td></tr><tr><td>s<sub>o,ac</sub></td><td>7/2</td><td>-</td><td>O<sub>2</sub> stoic. coef. during acetate consumption</td><td>(3)</td></tr><tr><td>s<sub>o,pro</sub></td><td>7/2</td><td>-</td><td>O<sub>2</sub> stoic. coef. during propionate consumption</td><td>(3)</td></tr><tr><td>s<sub>o,but</sub></td><td>7/2</td><td>-</td><td>O<sub>2</sub> stoic. coef. during butyrate consumption</td><td>(3)</td></tr><tr><td>s<sub>la</sub></td><td>1</td><td>-</td><td>Lactate stoic. coef.</td><td>(3)</td></tr><tr><td>s<sub>ac</sub></td><td>1</td><td>-</td><td>Acetate stoic. coef.</td><td>(3)</td></tr><tr><td>s<sub>pro</sub></td><td>1</td><td>-</td><td>Propionate stoic. coef.</td><td>(3)</td></tr><tr><td>s<sub>but</sub></td><td>1</td><td>-</td><td>Butyrate stoic. coef.</td><td>(3)</td></tr><tr><td>γ<sup>β</sup><sub>la</sub></td><td>0</td><td>10<sup>-15</sup> × <sup>mmol</sup>/<sub>h×μm<sup>2</sup></sub></td><td>Max. reaction speed</td><td>(3)</td></tr><tr><td>γ<sup>β</sup><sub>ac</sub></td><td>= γ<sup>β</sup><sub>la</sub></td><td>10<sup>-15</sup> × <sup>mmol</sup>/<sub>h×μm<sup>2</sup></sub></td><td>Max. reaction speed</td><td>(3)</td></tr><tr><td>γ<sup>β</sup><sub>pro</sub></td><td>= γ<sup>β</sup><sub>la</sub></td><td>10<sup>-15</sup> × <sup>mmol</sup>/<sub>h×μm<sup>2</sup></sub></td><td>Max. reaction speed</td><td>(3)</td></tr><tr><td>γ<sup>β</sup><sub>but</sub></td><td>2.2 × 10<sup>2</sup></td><td>10<sup>-15</sup> × <sup>mmol</sup>/<sub>h×μm<sup>2</sup></sub></td><td>Max. reaction speed</td><td>(3)</td></tr><tr><td>K<sub>la</sub></td><td>(2 × 10<sup>3</sup>)<sup>2/7</sup></td><td>mM</td><td>Affinity of β-oxidation</td><td>(3)</td></tr><tr><td>K<sub>ac</sub></td><td>(2 × 10<sup>3</sup>)<sup>2/7</sup></td><td>mM</td><td>Affinity of β-oxidation</td><td>(3)</td></tr><tr><td>K<sub>pro</sub></td><td>(2 × 10<sup>3</sup>)<sup>2/7</sup></td><td>mM</td><td>Affinity of β-oxidation</td><td>(3)</td></tr><tr><td>K<sub>but</sub></td><td>(9 × 10<sup>3</sup>)<sup>2/7</sup></td><td>mM</td><td>Affinity of β-oxidation</td><td>(3)</td></tr><tr><td>K<sub>HIF</sub></td><td>1.5</td><td>mM</td><td>Butyrate activation of β-oxid.</td><td>(3)</td></tr><tr><td>l<sub>HIF</sub></td><td>0.5</td><td>mM</td><td>Butyrate activation of β-oxid.</td><td>(3)</td></tr><tr><td>K<sub>H<sub>2</sub>S</sub></td><td>0.05</td><td>mM</td><td>H<sub>2</sub>S inhibition of β-oxid.</td><td>(3)</td></tr><tr><td>l<sub>H<sub>2</sub>S</sub></td><td>0.001</td><td>mM</td><td>H<sub>2</sub>S inhibition of β-oxid.</td><td>(3)</td></tr><tr><td colspan="5"><strong>Metabolites absorption parameters</strong></td></tr><tr><td>γ<sup>α</sup><sub>la</sub></td><td>70</td><td>h<sup>-1</sup></td><td>Lactate absorption rate</td><td>(3)</td></tr><tr><td>γ<sup>α</sup><sub>ac</sub></td><td>78</td><td>h<sup>-1</sup></td><td>Acetate absorption rate</td><td>(3)</td></tr><tr><td>γ<sup>α</sup><sub>pro</sub></td><td>95</td><td>h<sup>-1</sup></td><td>Propionate absorption rate</td><td>(3)</td></tr><tr><td>γ<sup>α</sup><sub>but</sub></td><td>70</td><td>h<sup>-1</sup></td><td>Butyrate absorption rate</td><td>(3)</td></tr><tr><td colspan="5"><strong>Concentration diffusion coefficients and boundary condition</strong></td></tr><tr><td>σ<sub>o</sub></td><td>3.6 × 10<sup>6</sup></td><td>μm<sup>2</sup>/h</td><td>O<sub>2</sub> diffusion coef.</td><td>(2)</td></tr><tr><td>σ<sub>la</sub></td><td>= σ<sub>o</sub></td><td>μm<sup>2</sup>/h</td><td>Lactate diffusion coef.</td><td>(3)</td></tr><tr><td>σ<sub>ac</sub></td><td>= σ<sub>o</sub></td><td>μm<sup>2</sup>/h</td><td>Acetate diffusion coef.</td><td>(3)</td></tr></tbody></table>
<table><caption>Table SM4 – Continued</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th><th>Refs</th></tr></thead><tbody><tr><td>σ<sub>pro</sub></td><td>= σ<sub>o</sub></td><td>μm<sup>2</sup>/h</td><td>Propionate diffusion coef.</td><td>(3)</td></tr><tr><td>σ<sub>but</sub></td><td>= σ<sub>o</sub></td><td>μm<sup>2</sup>/h</td><td>Butyrate diffusion coef.</td><td>(2)</td></tr><tr><td colspan="5"><strong>Cell interaction parameters</strong></td></tr><tr><td>W</td><td>6.01</td><td>-</td><td>Interaction force</td><td>-</td></tr><tr><td>D</td><td>8</td><td>-</td><td>Local density effect on cells</td><td>(3)</td></tr><tr><td colspan="5"><strong>DCS cell density shape</strong></td></tr><tr><td>N<sub>dcs</sub></td><td>36</td><td>-</td><td>Number of DCS cells</td><td>(3)</td></tr><tr><td>z<sub>d</sub></td><td>6</td><td>μm</td><td>Shape parameter</td><td>(3)</td></tr><tr><td>d</td><td>20</td><td>μm</td><td>Shape parameter</td><td>(3)</td></tr><tr><td>z<sub>u</sub></td><td>3</td><td>μm</td><td>Shape parameter</td><td>(3)</td></tr><tr><td>u</td><td>1/15</td><td>μm</td><td>Shape parameter</td><td>(3)</td></tr><tr><td colspan="5"><strong>Boundary conditions at the bottom of the crypt for densities</strong></td></tr><tr><td>ρ<sup>bot</sup><sub>sc</sub></td><td>0.6</td><td></td><td>Stem cell dentity</td><td>(3)</td></tr><tr><td>ρ<sup>bot</sup><sub>pc</sub></td><td>0</td><td></td><td>Progenitor cell dentity</td><td>(3)</td></tr><tr><td>ρ<sup>bot</sup><sub>gc</sub></td><td>0</td><td></td><td>Goblet cell dentity</td><td>(3)</td></tr><tr><td>ρ<sup>bot</sup><sub>ent</sub></td><td>0</td><td></td><td>Enterocyte cell dentity</td><td>(3)</td></tr><tr><td colspan="5"><strong>Boundary conditions at the bottom of the crypt for densities</strong></td></tr><tr><td>c̃<sup>bot</sup><sub>o</sub></td><td>48.8</td><td>[a.u]</td><td>Oxygen concentration</td><td>(3)</td></tr><tr><td colspan="5"><strong>Cell fate event regulation for stem cells (sc)</strong></td></tr><tr><td>q<sub>div,sc</sub></td><td>0.15</td><td>h<sup>-1</sup></td><td>Max. division rate of sc</td><td>(2)</td></tr><tr><td>K<sub>div,sc</sub>[z]</td><td>24</td><td>μm</td><td>Space-reg. for sc div.</td><td>(3)</td></tr><tr><td>κ<sub>div,sc</sub>[z]</td><td>5</td><td>μm</td><td>Space-related reg. for sc div.</td><td>(3)</td></tr><tr><td>K<sub>div,sc</sub>[dens]</td><td>50</td><td>cell</td><td>Density-reg. for sc div.</td><td>-</td></tr><tr><td>κ<sub>div,sc</sub>[dens]</td><td>6</td><td>cell</td><td>Density-related reg. for sc div.</td><td>(2)</td></tr><tr><td>K<sub>div,sc</sub>[but]</td><td>1</td><td>-</td><td>Butyrate-related reg. for sc div.</td><td>(3)</td></tr><tr><td>κ<sub>div,sc</sub>[but]</td><td>5</td><td>-</td><td>Butyrate-related reg. for sc div.</td><td>(3)</td></tr><tr><td>K<sub>div,sc</sub>[PRRs]</td><td>0.003</td><td>-</td><td>PRRs-related reg. for sc div.</td><td>(3)</td></tr><tr><td>κ<sub>div,sc</sub>[PRRs]</td><td>0.003</td><td>-</td><td>PRRs-related reg. for sc div.</td><td>(3)</td></tr><tr><td>q<sub>sc,pc</sub></td><td>0.2</td><td>h<sup>-1</sup></td><td>Mac. differentiation rate of sc</td><td>(2)</td></tr><tr><td>K<sub>sc,pc</sub>[z]</td><td>= K<sub>div,sc</sub>[z]</td><td>μm</td><td>Space-reg. for sc diff.</td><td>(3)</td></tr><tr><td>κ<sub>sc,pc</sub>[z]</td><td>= κ<sub>div,sc</sub>[z]</td><td>μm</td><td>Space-related reg. for sc diff.</td><td>-</td></tr><tr><td colspan="5"><strong>Cell fate event regulation for progenitor cells (pc)</strong></td></tr><tr><td>q<sub>div,pc</sub></td><td>0.22</td><td>h<sup>-1</sup></td><td>Max division rate of pc cells</td><td>(3)</td></tr><tr><td>K<sub>div,pc</sub>[z]</td><td>110</td><td>μm</td><td>Space-reg. for pc div.</td><td>(3)</td></tr></tbody></table>
<table><caption>Table SM4 – Continued</caption><thead><tr><th>Parameter</th><th>Value</th><th>Unit</th><th>Description</th><th>Refs</th></tr></thead><tbody><tr><td>κ<sub>div,pc</sub>[z]</td><td>40</td><td>μm</td><td>Space-related reg. for pc div.</td><td>(3)</td></tr><tr><td>K<sub>div,pc</sub>[dens]</td><td>44</td><td>cell</td><td>Density-reg. for pc div.</td><td>-</td></tr><tr><td>κ<sub>div,pc</sub>[dens] = κ<sub>div,sc</sub>[dens]</td><td>cell</td><td>cell</td><td>Density-related reg. for pc div.</td><td>(2)</td></tr><tr><td>K<sub>div,pc</sub>[PRRs] = K<sub>div,sc</sub>[PRRs]</td><td>-</td><td>-</td><td>PRRs-related reg. for pc div.</td><td>(3)</td></tr><tr><td>κ<sub>div,pc</sub>[PRRs] = κ<sub>div,sc</sub>[PRRs]</td><td>-</td><td>-</td><td>PRRs-related reg. for pc div.</td><td>(3)</td></tr><tr><td>q<sub>pc,gc</sub> = q<sub>pc,ent</sub>/3</td><td>h<sup>-1</sup></td><td>h<sup>-1</sup></td><td>Max. differentiation rate of pc in goblet cells</td><td>(2)</td></tr><tr><td>K<sub>pc,gc</sub>[z] = K<sub>div,pc</sub>[z]</td><td>μm</td><td>μm</td><td>Space-reg. for pc diff. in gc</td><td>(3)</td></tr><tr><td>κ<sub>pc,gc</sub>[z] = 15</td><td>μm</td><td>μm</td><td>Space-reg. for pc diff. in gc</td><td>(3)</td></tr><tr><td>K<sub>pc,gc</sub>[but] = 0.8</td><td>-</td><td>-</td><td>Butyrate-related reg. for pc diff. in gc</td><td>(3)</td></tr><tr><td>κ<sub>pc,gc</sub>[but] = 0.5</td><td>-</td><td>-</td><td>Butyrate-related reg. for pc diff. in gc</td><td>(3)</td></tr><tr><td>q<sub>pc,ent</sub> = 0.28</td><td>h<sup>-1</sup></td><td>h<sup>-1</sup></td><td>Max diff. rate of pc in enterocyte</td><td>-</td></tr><tr><td>K<sub>pc,ent</sub>[z] = K<sub>div,pc</sub>[z]</td><td>μm</td><td>μm</td><td>Space-reg. for pc diff. in ent</td><td>(3)</td></tr><tr><td>κ<sub>pc,ent</sub>[z] = 30</td><td>μm</td><td>μm</td><td>Space-reg. for pc diff. in ent</td><td>-</td></tr><tr><td>K<sub>pc,ent</sub>[but] = K<sub>pc,gc</sub>[but]</td><td>-</td><td>-</td><td>Butyrate-related reg. for pc diff. in ent</td><td>(3)</td></tr><tr><td>κ<sub>pc,ent</sub>[but] = κ<sub>pc,gc</sub>[but]</td><td>-</td><td>-</td><td>Butyrate-related reg. for pc diff. in ent</td><td>(3)</td></tr><tr><td colspan="5"><strong>Cell fate event regulation for goblet cells (gc)</strong></td></tr><tr><td>q<sub>ex,gc</sub> = 0.34</td><td>h<sup>-1</sup></td><td>h<sup>-1</sup></td><td>Max rate of extrusion</td><td>(2)</td></tr><tr><td>K<sub>ex,gc</sub>[z] = 385</td><td>μm</td><td>μm</td><td>Space-reg. for gc ext.</td><td>-</td></tr><tr><td>κ<sub>ex,gc</sub>[z] = 15</td><td>μm</td><td>μm</td><td>Space-related reg. for gc ext.</td><td>(3)</td></tr><tr><td>K<sub>ex,gc</sub>[dens] = 41</td><td>cell</td><td>cell</td><td>Density-reg. for gc ext.</td><td>-</td></tr><tr><td>κ<sub>ex,gc</sub>[dens] = κ<sub>div,sc</sub>[dens]</td><td>cell</td><td>cell</td><td>Density-related reg. for sc div.</td><td>(2)</td></tr><tr><td colspan="5"><strong>Cell fate event regulation for enterocyte cells (ent)</strong></td></tr><tr><td>q<sub>ex,gc</sub> = q<sub>ex,gc</sub></td><td>h<sup>-1</sup></td><td>h<sup>-1</sup></td><td>Max rate of extrusion</td><td>(2)</td></tr><tr><td>K<sub>ex,ent</sub>[z] = K<sub>ex,ent</sub>[z]</td><td>μm</td><td>μm</td><td>Space-reg. for gc ext.</td><td>(3)</td></tr><tr><td>κ<sub>ex,ent</sub>[z] = κ<sub>ex,gc</sub>[z]</td><td>μm</td><td>μm</td><td>Space-related reg. for gc ext.</td><td>(3)</td></tr><tr><td>K<sub>ex,ent</sub>[dens] = K<sub>ex,gc</sub>[dens]</td><td>cell</td><td>cell</td><td>Density-reg. for gc ext.</td><td>(2)</td></tr><tr><td>κ<sub>ex,ent</sub>[dens] = κ<sub>div,sc</sub>[dens]</td><td>cell</td><td>cell</td><td>Density-related reg. for sc div.</td><td>(2)</td></tr></tbody></table>
**SM2 Biological and physiological processes** The biochemical dynamics of the model are governed by a set of 14 processes (denoted as *P1*, *P2*,...), describing the interaction between microbial functional groups and metabolic substrates. Figure SM1 provides a schematic overview of these interactions, while their kinetic parameters and yield coefficients are detailed in Section SM1.

<figure><img src="image_16.png" /><figcaption>Fig. SM1: **Metabolic and regulatory processes representation** (Figure adapted from (3)). This diagram provides an overview of the key interactions among different microbial communities involved in polysaccharide and protein degradation. Metabolic pathways are uniquely labeled by color and number, with arrows tracing the transformation from substrates to final products and linking them to the associated microbial groups. Oxygen consumption (+O₂) is used to indicate oxidative reactions. Metabolites with similar functions are grouped in the same box. The figure also captures gas-phase transitions. Regulatory elements include the inhibitory effect of H₂S on mucus-associated processes. The model integrates death rates modulated by baseline activity, antimicrobial peptides (AMPs) and local oxygen levels—used here as a marker for inflammation. Microbial responses are stratified into three sensitivity tiers: high (Φ), intermediate (η), and low (δ). A summary of the eight microbial groups present in the model is described in the corresponding table.</figcaption></figure>
**SM3 Process matrices** Let $P_f$ (respectively $P_c$) denote the process reaction matrix associated with solid components (respectively dissolved compounds). These matrices specify the component yields (see Table SM3) for each process included in the model. The source term vectors $F = (F_i)_{i \in I_S}$ and $G = (G_j)_{j \in I_D}$ are defined as

$$F = P_f \cdot K_f \quad \text{and} \quad G = P_c \cdot K_c,$$

where $K_f$ and $K_c$ denote the vectors of kinetic rates (3). Once the process matrices are defined, the source terms can be readily computed and used in the numerical simulations.

<table><caption>Table SM5: Process matrix $(P_f)^t$ and kinetic rate vector $K_f$.</caption><thead><tr><th></th><th>hm</th><th>sm</th><th>pol</th><th>prot</th><th>r</th><th>B<sup>0</sup><sub>man</sub></th><th>B<sup>0</sup><sub>sm</sub></th><th>B<sup>0</sup><sub>la</sub></th><th>B<sup>0</sup><sub>in</sub></th><th>B<sup>1</sup><sub>in</sub></th><th>B<sup>1</sup><sub>H2S</sub></th><th>B<sup>0</sup><sub>H2a</sub></th><th>B<sup>0</sup><sub>H2m</sub></th><th>l</th><th>Kinetic K<sub>f</sub></th></tr></thead><tbody><tr><td>P1<sup>0</sup><sub>sm</sub></td><td></td><td>-Y<sup>P1</sup><sub>sm</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P1</sup><sub>sm</sub></td><td>k<sup>P1</sup><sub>0.05J<sub>in</sub></sub>f<sub>B<sub>2a</sub></sub><sup>P1</sup><sub>0.05f<sub>in</sub></sub></td></tr><tr><td>P1<sup>1</sup><sub>sm</sub></td><td></td><td>-Y<sup>P1</sup><sub>sm</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P1</sup><sub>sm</sub></td><td>k<sup>P1</sup><sub>0.05J<sub>in</sub></sub>f<sub>B<sub>2a</sub></sub><sup>P1</sup><sub>0.05f<sub>in</sub></sub></td></tr><tr><td>P1<sup>0</sup><sub>pol</sub></td><td></td><td></td><td>-Y<sup>P1</sup><sub>pol</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P1</sup><sub>pol</sub></td><td>k<sup>P1</sup><sub>0.05J<sub>in</sub></sub>f<sub>B<sub>2a</sub></sub><sup>P1</sup><sub>0.05f<sub>in</sub></sub></td></tr><tr><td>P2<sub>sm</sub></td><td></td><td>-Y<sup>P2</sup><sub>sm</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P2</sup><sub>B<sub>H2a</sub></sub></td><td>Y<sup>P2</sup><sub>sm</sub> - Y<sup>P2</sup><sub>B<sub>H2a</sub></sub></td><td>k<sup>P2</sup><sub>0.05J<sub>in</sub></sub>f<sub>B<sub>2a</sub></sub><sup>P2</sup><sub>0.05f<sub>in</sub></sub></td></tr><tr><td>P2<sub>prot</sub></td><td></td><td></td><td>-Y<sup>P2</sup><sub>prot</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P2</sup><sub>B<sub>H2a</sub></sub></td><td>Y<sup>P2</sup><sub>prot</sub> - Y<sup>P2</sup><sub>B<sub>H2a</sub></sub></td><td>k<sup>P2</sup><sub>0.05J<sub>in</sub></sub>f<sub>B<sub>H2a</sub></sub><sup>P2</sup><sub>0.05f<sub>in</sub></sub></td></tr><tr><td>P3<sup>0</sup></td><td></td><td></td><td></td><td>Y<sup>P3</sup><sub>B<sub>man</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-Y<sup>P3</sup><sub>B<sub>man</sub></sub></td><td>k<sup>P3</sup><sub>0.05J<sub>in</sub></sub>f<sub>B<sub>2a</sub></sub><sup>P3</sup><sub>0.05f<sub>in</sub></sub></td></tr><tr><td>P3<sup>1</sup></td><td></td><td></td><td></td><td></td><td>Y<sup>P3</sup><sub>B<sub>man</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-Y<sup>P3</sup><sub>B<sub>man</sub></sub></td><td>k<sup>P3</sup><sub>0.05J<sub>in</sub></sub>f<sub>B<sub>2a</sub></sub><sup>P3</sup><sub>0.05f<sub>in</sub></sub></td></tr><tr><td>P4<sup>0</sup></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P4</sup><sub>B<sub>in</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-Y<sup>P4</sup><sub>B<sub>in</sub></sub></td><td>k<sup>P4</sup><sub>0.05J<sub>in</sub></sub>f<sub>B<sub>2a</sub></sub><sup>P4</sup><sub>0.05f<sub>in</sub></sub></td></tr><tr><td>P4<sup>1</sup></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P4</sup><sub>B<sub>in</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td>-Y<sup>P4</sup><sub>B<sub>in</sub></sub></td><td>k<sup>P4</sup><sub>0.05J<sub>in</sub></sub>f<sub>B<sub>2a</sub></sub><sup>P4</sup><sub>0.05f<sub>in</sub></sub></td></tr><tr><td>P4<sup>4</sup></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P4</sup><sub>B<sub>in</sub></sub></td><td></td><td></td><td></td><td></td><td>-Y<sup>P4</sup><sub>B<sub>in</sub></sub></td><td>k<sup>P4</sup><sub>0.05J<sub>in</sub></sub>f<sub>B<sub>2a</sub></sub><sup>P4</sup><sub>0.05f<sub>in</sub></sub></td></tr><tr><td>P5</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P5</sup><sub>B<sub>H2a</sub></sub></td><td></td><td></td><td></td><td>Y<sup>P5</sup><sub>B<sub>H2a</sub></sub></td><td>k<sup>P5</sup><sub>0.05J<sub>in</sub></sub>f<sub>B<sub>2a</sub></sub><sup>P5</sup><sub>0.05f<sub>in</sub></sub></td></tr><tr><td>P6</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P6</sup><sub>B<sub>H2a</sub></sub></td><td></td><td></td><td>-Y<sup>P6</sup><sub>B<sub>H2a</sub></sub></td><td>k<sup>P6</sup><sub>0.05J<sub>in</sub></sub>f<sub>B<sub>2a</sub></sub><sup>P6</sup><sub>0.05f<sub>in</sub></sub></td></tr><tr><td>P7</td><td></td><td>-Y<sup>P7</sup><sub>sm</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P7</sup><sub>sm</sub></td><td>k<sup>P7</sup>R(c<sub>H2a</sub>, K<sup>P7</sup>, K<sup>P7</sup>)f<sub>in</sub></td></tr><tr><td>P8<sup>0</sup><sub>man</sub></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P8</sup>R(c<sub>H2a</sub>, K<sup>P8</sup>, K<sup>P8</sup>)f<sub>B<sub>H2a</sub></sub><sup>P8</sup></td></tr><tr><td>P8<sup>1</sup><sub>man</sub></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P8</sup>R(c<sub>H2a</sub>, K<sup>P8</sup>, K<sup>P8</sup>)f<sub>B<sub>H2a</sub></sub><sup>P8</sup></td></tr><tr><td>P8<sup>0</sup><sub>la</sub></td><td></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P8</sup>R(c<sub>H2a</sub>, K<sup>P8</sup>, K<sup>P8</sup>)f<sub>B<sub>in</sub></sub><sup>P8</sup></td></tr><tr><td>P8<sup>1</sup><sub>la</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P8</sup>R(c<sub>H2a</sub>, K<sup>P8</sup>, K<sup>P8</sup>)f<sub>B<sub>in</sub></sub><sup>P8</sup></td></tr><tr><td>P8<sup>0</sup><sub>B<sub>H2a</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P8</sup>R(c<sub>H2a</sub>, K<sup>P8</sup>, K<sup>P8</sup>)f<sub>B<sub>H2a</sub></sub><sup>P8</sup></td></tr><tr><td>P8<sup>1</sup><sub>B<sub>H2a</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P8</sup>R(c<sub>H2a</sub>, K<sup>P8</sup>, K<sup>P8</sup>)f<sub>B<sub>H2a</sub></sub><sup>P8</sup></td></tr><tr><td>P9<sup>0</sup><sub>man</sub></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P9</sup>R(c<sub>AMP<sub>La</sub></sub>, K<sup>P9</sup>, K<sup>P9</sup>)f<sub>B<sub>H2a</sub></sub><sup>P9</sup></td></tr><tr><td>P9<sup>1</sup><sub>man</sub></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P9</sup>R(c<sub>AMP<sub>La</sub></sub>, K<sup>P9</sup>, K<sup>P9</sup>)f<sub>B<sub>H2a</sub></sub><sup>P9</sup></td></tr><tr><td>P9<sup>0</sup><sub>B<sub>in</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P9</sup>R(c<sub>AMP<sub>La</sub></sub>, K<sup>P9</sup>, K<sup>P9</sup>)f<sub>B<sub>in</sub></sub><sup>P9</sup></td></tr><tr><td>P9<sup>1</sup><sub>B<sub>in</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P9</sup>R(c<sub>AMP<sub>La</sub></sub>, K<sup>P9</sup>, K<sup>P9</sup>)f<sub>B<sub>in</sub></sub><sup>P9</sup></td></tr><tr><td>P9<sup>0</sup><sub>B<sub>H2a</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P9</sup>R(c<sub>AMP<sub>La</sub></sub>, K<sup>P9</sup>, K<sup>P9</sup>)f<sub>B<sub>H2a</sub></sub><sup>P9</sup></td></tr><tr><td>P9<sup>1</sup><sub>B<sub>H2a</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P9</sup>R(c<sub>AMP<sub>La</sub></sub>, K<sup>P9</sup>, K<sup>P9</sup>)f<sub>B<sub>H2a</sub></sub><sup>P9</sup></td></tr><tr><td>P9<sup>0</sup><sub>B<sub>H2a</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P9</sup>R(c<sub>AMP<sub>La</sub></sub>, K<sup>P9</sup>, K<sup>P9</sup>)f<sub>B<sub>H2a</sub></sub><sup>P9</sup></td></tr><tr><td>P9<sup>1</sup><sub>B<sub>H2a</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P9</sup>R(c<sub>AMP<sub>La</sub></sub>, K<sup>P9</sup>, K<sup>P9</sup>)f<sub>B<sub>H2a</sub></sub><sup>P9</sup></td></tr><tr><td>P10<sup>0</sup><sub>man</sub></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P10</sup>f<sub>B<sub>man</sub></sub><sup>P10</sup></td></tr><tr><td>P10<sup>1</sup><sub>man</sub></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P10</sup>f<sub>B<sub>man</sub></sub><sup>P10</sup></td></tr><tr><td>P10<sup>0</sup><sub>B<sub>in</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P10</sup>f<sub>B<sub>in</sub></sub><sup>P10</sup></td></tr><tr><td>P10<sup>1</sup><sub>B<sub>in</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P10</sup>f<sub>B<sub>in</sub></sub><sup>P10</sup></td></tr><tr><td>P10<sup>0</sup><sub>B<sub>H2a</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P10</sup>f<sub>B<sub>H2a</sub></sub><sup>P10</sup></td></tr><tr><td>P10<sup>1</sup><sub>B<sub>H2a</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>k<sup>P10</sup>f<sub>B<sub>H2a</sub></sub><sup>P10</sup></td></tr><tr><td>P12</td><td></td><td>-Y<sup>P12</sup><sub>man</sub></td><td>Y<sup>P12</sup><sub>sm</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P12</sup><sub>man</sub> - Y<sup>P12</sup><sub>sm</sub></td><td>k<sup>P12</sup>f<sub>man</sub><sup>P12</sup></td></tr></tbody></table>
<table><caption>Table SM6: Process matrix ($P_c$)$^t$ and kinetic rate vector $K_c$.</caption><thead><tr><th>Process</th><th>mon</th><th>la</th><th>ac</th><th>pro</th><th>but</th><th>AMPs</th><th>H<sub>2</sub>S</th><th>CH<sub>4</sub></th><th>CO<sub>2</sub></th><th>H<sub>2</sub></th><th>O<sub>2</sub></th><th>Kinetic K<sub>c</sub></th></tr></thead><tbody><tr><td>P1<sup>1</sup><sub>mon</sub></td><td>Y<sup>P1</sup><sub>mon</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>k<sup>P1</sup><sup>0.05f<sub>mon</sub>f<sub>mon</sub><sup>2</sup></sup><sup>K<sub>f</sub><sup>P1</sup></sup></td></tr><tr><td>P1<sup>1</sup><sub>mon</sub></td><td>Y<sup>P1</sup><sub>mon</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>k<sup>P1</sup><sup>0.05f<sub>mon</sub>f<sub>mon</sub><sup>2</sup></sup><sup>K<sub>f</sub><sup>P1</sup></sup></td></tr><tr><td>P1<sup>1</sup><sub>pol</sub></td><td>Y<sup>P1</sup><sub>pol</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>k<sup>P1</sup><sup>0.05f<sub>mon</sub>f<sub>mon</sub><sup>2</sup></sup><sup>K<sub>f</sub><sup>P1</sup></sup></td></tr><tr><td>P1<sup>1</sup><sub>pol</sub></td><td>Y<sup>P1</sup><sub>pol</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>k<sup>P1</sup><sup>0.05f<sub>mon</sub>f<sub>mon</sub><sup>2</sup></sup><sup>K<sub>f</sub><sup>P1</sup></sup></td></tr><tr><td>P2<sub>mon</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P2<sub>mon</sub></sup><sub>H<sub>2</sub>S</sub></td><td></td><td>Y<sup>P2</sup><sub>CO<sub>2</sub></sub></td><td>Y<sup>P2</sup><sub>H<sub>2</sub></sub></td><td>Y<sup>P2</sup><sub>O<sub>2</sub></sub></td><td></td></tr><tr><td>P2<sub>prot</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P2<sub>prot</sub></sup><sub>H<sub>2</sub>S</sub></td><td></td><td>Y<sup>P2</sup><sub>CO<sub>2</sub></sub></td><td>Y<sup>P2</sup><sub>H<sub>2</sub></sub></td><td>Y<sup>P2</sup><sub>O<sub>2</sub></sub></td><td></td></tr><tr><td>P3<sup>1</sup></td><td>-1</td><td>Y<sup>P3</sup><sub>la</sub></td><td>Y<sup>P3</sup><sub>ac</sub></td><td>Y<sup>P3</sup><sub>pro</sub></td><td>Y<sup>P3</sup><sub>but</sub></td><td></td><td></td><td></td><td>Y<sup>P3</sup><sub>CO<sub>2</sub></sub></td><td>Y<sup>P3</sup><sub>H<sub>2</sub></sub></td><td></td><td>k<sup>P2</sup><sup>f<sub>mon</sub><sup>2</sup>f<sub>mon</sub><sup>2</sup></sup><sup>f<sub>mon</sub><sup>2</sup></sup></td></tr><tr><td>P3<sup>1</sup></td><td>-1</td><td>Y<sup>P3</sup><sub>la</sub></td><td>Y<sup>P3</sup><sub>ac</sub></td><td>Y<sup>P3</sup><sub>pro</sub></td><td>Y<sup>P3</sup><sub>but</sub></td><td></td><td></td><td></td><td>Y<sup>P3</sup><sub>CO<sub>2</sub></sub></td><td>Y<sup>P3</sup><sub>H<sub>2</sub></sub></td><td></td><td>k<sup>P2</sup><sup>f<sub>mon</sub><sup>2</sup>f<sub>mon</sub><sup>2</sup></sup><sup>f<sub>mon</sub><sup>2</sup></sup></td></tr><tr><td>P4<sup>1</sup></td><td>-1</td><td>Y<sup>P4</sup><sub>ac</sub></td><td>Y<sup>P4</sup><sub>pro</sub></td><td>Y<sup>P4</sup><sub>but</sub></td><td></td><td></td><td></td><td></td><td>Y<sup>P4</sup><sub>CO<sub>2</sub></sub></td><td>Y<sup>P4</sup><sub>H<sub>2</sub></sub></td><td></td><td>k<sup>P2</sup><sup>f<sub>mon</sub><sup>2</sup>f<sub>mon</sub><sup>2</sup></sup><sup>f<sub>mon</sub><sup>2</sup></sup></td></tr><tr><td>P4<sup>1</sup></td><td>-1</td><td>Y<sup>P4</sup><sub>ac</sub></td><td>Y<sup>P4</sup><sub>pro</sub></td><td>Y<sup>P4</sup><sub>but</sub></td><td></td><td></td><td></td><td></td><td>Y<sup>P4</sup><sub>CO<sub>2</sub></sub></td><td>Y<sup>P4</sup><sub>H<sub>2</sub></sub></td><td></td><td>k<sup>P3</sup><sup>f<sub>mon</sub><sup>2</sup>f<sub>mon</sub><sup>2</sup></sup><sup>f<sub>mon</sub><sup>2</sup></sup></td></tr><tr><td>P4<sup>1</sup></td><td>-1</td><td>Y<sup>P4</sup><sub>ac</sub></td><td>Y<sup>P4</sup><sub>pro</sub></td><td>Y<sup>P4</sup><sub>but</sub></td><td></td><td></td><td></td><td></td><td>Y<sup>P4</sup><sub>CO<sub>2</sub></sub></td><td>Y<sup>P4</sup><sub>H<sub>2</sub></sub></td><td></td><td>k<sup>P3</sup><sup>f<sub>mon</sub><sup>2</sup>f<sub>mon</sub><sup>2</sup></sup><sup>f<sub>mon</sub><sup>2</sup></sup></td></tr><tr><td>P5</td><td></td><td>Y<sup>P5</sup><sub>ac</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P5</sup><sub>CO<sub>2</sub></sub></td><td>-1</td><td></td></tr><tr><td>P6</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>Y<sup>P6</sup><sub>CH<sub>4</sub></sub></td><td>-1</td><td></td></tr><tr><td>P11<sub>H<sub>2</sub>S</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td></td><td>k<sup>P11</sup>(c<sub>H<sub>2</sub>S</sub> - K<sup>P11</sup><sub>H<sub>2</sub></sub>RT[H<sub>2</sub>S<sub>g</sub>]<sub>∞</sub>)</td></tr><tr><td>P11<sub>CH<sub>4</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td></td><td>k<sup>P11</sup>(c<sub>CH<sub>4</sub></sub> - K<sup>P11</sup><sub>CH<sub>4</sub></sub>RT[CH<sub>4</sub>,g]<sub>∞</sub>)</td></tr><tr><td>P11<sub>CO<sub>2</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td></td><td>k<sup>P11</sup>(c<sub>CO<sub>2</sub></sub> - K<sup>P11</sup><sub>CO<sub>2</sub></sub>RT[CO<sub>2</sub>,g]<sub>∞</sub>)</td></tr><tr><td>P11<sub>H<sub>2</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-1</td><td></td><td>k<sup>P11</sup>(c<sub>H<sub>2</sub></sub> - K<sup>P11</sup><sub>H<sub>2</sub></sub>RT[H<sub>2</sub>,g]<sub>∞</sub>)</td></tr><tr><td>P13<sub>O<sub>2</sub></sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-Y<sup>P13</sup><sub>O<sub>2</sub></sub></td><td>k<sup>P13</sup><sub>CO<sub>2</sub></sub></td></tr><tr><td>P14<sub>AMPs</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-Y<sup>P14</sup><sub>AMPs</sub></td><td></td><td></td><td></td><td></td><td>k<sup>P14</sup><sub>AMPs</sub></td></tr></tbody></table>
## SM4 Source Terms

### SM4.1 Solid components:

$$ 
\begin{aligned}
F_{sm} = & Y_{sm}^{P12} k^{P12} f_{hm} - Y_{sm}^{P1} k^{P1} \frac{0.05 f_{sm} f_{B_{mon}^{\phi}} K_{f}^{P1}}{K_{f}^{P1} f_{B_{mon}^{\phi}} + 0.05 f_{sm}} + \\ & - Y_{sm}^{P1} k^{P1} \frac{0.05 f_{sm} f_{B_{mon}^{\eta}} K_{f}^{P1}}{K_{f}^{P1} f_{B_{mon}^{\eta}} + 0.05 f_{sm}} - Y_{sm}^{P2} k^{P2} f_{l} \frac{0.05 f_{sm} f_{B_{H2S}^{\delta}}}{K_{f}^{P2} f_{B_{H2S}^{\delta}} + 0.05 f_{sm}} + \\ & - Y_{sm}^{P7} k^{P7} R(c_{H2S}, K^{P7}, \kappa^{P7}) 0.05 f_{sm}
\end{aligned}
$$

$$ F_{pol} = -Y_{pol}^{P1} k^{P1} \frac{0.05 f_{sm} f_{B_{mon}^{\phi}} K_{f}^{P1}}{K_{f}^{P1} f_{B_{mon}^{\phi}} + 0.05 f_{sm}} - Y_{pol}^{P1} k^{P1} \frac{0.05 f_{sm} f_{B_{mon}^{\eta}} K_{f}^{P1}}{K_{f}^{P1} f_{B_{mon}^{\eta}} + 0.05 f_{sm}} $$

$$ F_{prot} = -Y_{prot}^{P2} k^{P2} f_{prot} f_{l} \frac{f_{prot} f_{B_{H2S}^{\delta}}}{K_{f}^{P2} f_{B_{H2S}^{\delta}} + f_{prot}} $$

$$ F_r = 0 $$

$$ F_{B_{mon}^{\phi}} = Y_{B_{mon}^{P3}} k^{P3} f_{l} \frac{c_{mon} f_{B_{mon}^{\phi}}}{K_{s}^{P3} + c_{mon}} - k^{P8,\phi} R(c_{O2}, K^{P8,\phi}, \kappa^{P8,\phi}) f_{B_{mon}^{\phi}} + \\ - k^{P9} R(c_{AMPs}, K^{P9}, \kappa^{P9}) f_{B_{mon}^{\phi}} - k^{P10} f_{B_{mon}^{\phi}} $$

$$ F_{B_{mon}^{\eta}} = Y_{B_{mon}^{P3}} k^{P3} f_{l} \frac{c_{mon} f_{B_{mon}^{\eta}}}{K_{s}^{P3} + c_{mon}} - k^{P8,\eta} R(c_{O2}, K^{P8,\eta}, \kappa^{P8,\eta}) f_{B_{mon}^{\eta}} + \\ - k^{P9} R(c_{AMPs}, K^{P9}, \kappa^{P9}) f_{B_{mon}^{\eta}} - k^{P10} f_{B_{mon}^{\eta}} $$

$$ F_{B_{la}^{\phi}} = Y_{B_{la}^{P4}} k^{P4} f_{l} \frac{c_{la} f_{B_{la}^{\phi}}}{K_{s}^{P4} + c_{la}} - k^{P8,\phi} R(c_{O2}, K^{P8,\phi}, \kappa^{P8,\phi}) f_{B_{la}^{\phi}} + \\ - k^{P9} R(c_{AMPs}, K^{P9}, \kappa^{P9}) f_{B_{la}^{\phi}} - k^{P10} f_{B_{la}^{\phi}} $$

$$ F_{B_{la}^{\eta}} = Y_{B_{la}^{P4}} k^{P4} f_{l} \frac{c_{la} f_{B_{la}^{\eta}}}{K_{s}^{P4} + c_{la}} - k^{P8,\eta} R(c_{O2}, K^{P8,\eta}, \kappa^{P8,\eta}) f_{B_{la}^{\eta}} + \\ - k^{P9} R(c_{AMPs}, K^{P9}, \kappa^{P9}) f_{B_{la}^{\eta}} - k^{P10} f_{B_{la}^{\eta}} $$

$$ F_{B_{la}^{\delta}} = Y_{B_{la}^{P4}} k^{P4} f_{l} \frac{c_{la} f_{B_{la}^{\delta}}}{K_{s}^{P4} + c_{la}} - k^{P9} R(c_{AMPs}, K^{P9}, \kappa^{P9}) f_{B_{la}^{\delta}} - k^{P10} f_{B_{la}^{\delta}} $$

$$ F_{B_{H2S}} = Y_{B_{H2S}^{P2m}} k^{P2m} f_{l} \frac{f_{sm} f_{B_{H2S}^{\delta}}}{K_{f}^{P2} f_{B_{H2S}^{\delta}} + f_{sm}} + Y_{B_{H2S}^{P2prot}} k^{P2prot} f_{l} \frac{f_{prot} f_{B_{H2S}^{\delta}}}{K_{f}^{P2} f_{B_{H2S}^{\delta}} + f_{prot}} + \\ - k^{P9} R(c_{AMPs}, K^{P9}, \kappa^{P9}) f_{B_{H2S}^{\delta}} - k^{P10} f_{B_{H2S}^{\delta}} $$

$$ F_{B_{H2a}} = Y_{B_{H2a}^{\phi}} k^{P5} f_{l} \frac{c_{H2} f_{B_{H2a}^{\phi}}}{K_{s}^{P5} + c_{H2}} - k^{P8,\phi} R(c_{O2}, K^{P8,\phi}, \kappa^{P8,\phi}) f_{B_{H2a}^{\phi}} + \\ - k^{P9} R(c_{AMPs}, K^{P9}, \kappa^{P9}) f_{B_{H2a}^{\phi}} - k^{P10} f_{B_{H2a}^{\phi}} $$

$$ F_{B_{H2m}} = Y_{B_{H2m}^{\phi}} k^{P6} f_{l} \frac{c_{H2} f_{B_{H2m}^{\phi}}}{K_{s}^{P6} + c_{H2}} I_{pH} - k^{P8,\phi} R(c_{O2}, K^{P8,\phi}, \kappa^{P8,\phi}) f_{B_{H2m}^{\phi}} + $$
$$ -k^{P9}R(c_{AMPs}, K^{P9}, \kappa^{P9})f_{B_{H2m}^{\phi}} - k^{P10}f_{B_{H2m}^{\phi}} $$

$$ F_{hm} = -Y_{hm}^{P12}k^{P12}f_{hm} $$

## SM4.2 Dissolved compounds

$$ G_{mon} = Y_{mon}^{P1}k^{P1} \frac{0.05f_{sm}f_{B_{mon}^{\phi}}K_{f}^{P1}}{K_{f}^{P1}f_{B_{mon}^{\phi}} + 0.05f_{sm}} + Y_{mon}^{P1}k^{P1} \frac{0.05f_{sm}f_{B_{mon}^{\eta}}K_{f}^{P1}}{K_{f}^{P1}f_{B_{mon}^{\eta}} + 0.05f_{sm}} + \\ + Y_{pol}^{P1}k^{P1} \frac{0.05f_{sm}f_{B_{mon}^{\phi}}K_{f}^{P1}}{K_{f}^{P1}f_{B_{mon}^{\phi}} + 0.05f_{sm}} + Y_{pol}^{P1}k^{P1} \frac{0.05f_{sm}f_{B_{mon}^{\eta}}K_{f}^{P1}}{K_{f}^{P1}f_{B_{mon}^{\eta}} + 0.05f_{sm}} + \\ - k^{P3}f_{l} \frac{c_{mon}f_{B_{mon}^{\phi}}}{K_{s}^{P3} + c_{mon}} - k^{P3}f_{l} \frac{c_{mon}f_{B_{mon}^{\eta}}}{K_{s}^{P3} + c_{mon}} $$

$$ G_{la} = Y_{la}^{P3}k^{P3}f_{l} \frac{c_{mon}f_{B_{mon}^{\phi}}}{K_{s}^{P3} + c_{mon}} + Y_{la}^{P3}k^{P3}f_{l} \frac{c_{mon}f_{B_{mon}^{\eta}}}{K_{s}^{P3} + c_{mon}} - k^{P4}f_{l} \frac{c_{la}f_{B_{la}^{\phi}}}{K_{s}^{P4} + c_{la}} + \\ - k^{P4}f_{l} \frac{c_{la}f_{B_{la}^{\eta}}}{K_{s}^{P4} + c_{la}} - k^{P4}f_{l} \frac{c_{la}f_{B_{la}^{\delta}}}{K_{s}^{P4} + c_{la}} $$

$$ G_{ac} = Y_{ac}^{P3}k^{P3}f_{l} \frac{c_{mon}f_{B_{mon}^{\phi}}}{K_{s}^{P3} + c_{mon}} + Y_{ac}^{P3}k^{P3}f_{l} \frac{c_{mon}f_{B_{mon}^{\eta}}}{K_{s}^{P3} + c_{mon}} + Y_{ac}^{P4}k^{P4}f_{l} \frac{c_{la}f_{B_{la}^{\phi}}}{K_{s}^{P4} + c_{la}} + \\ + Y_{ac}^{P4}k^{P4}f_{l} \frac{c_{la}f_{B_{la}^{\eta}}}{K_{s}^{P4} + c_{la}} + Y_{ac}^{P4}k^{P4}f_{l} \frac{c_{la}f_{B_{la}^{\delta}}}{K_{s}^{P4} + c_{la}} + Y_{ac}^{P5}k^{P5}f_{l} \frac{c_{H_{2}}f_{B_{H2a}^{\phi}}}{K_{s}^{P5} + c_{H_{2}}} $$

$$ G_{pro} = Y_{pro}^{P3}k^{P3}f_{l} \frac{c_{mon}f_{B_{mon}^{\phi}}}{K_{s}^{P3} + c_{mon}} + Y_{pro}^{P3}k^{P3}f_{l} \frac{c_{mon}f_{B_{mon}^{\eta}}}{K_{s}^{P3} + c_{mon}} + Y_{pro}^{P4}k^{P4}f_{l} \frac{c_{la}f_{B_{la}^{\phi}}}{K_{s}^{P4} + c_{la}} + \\ + Y_{pro}^{P4}k^{P4}f_{l} \frac{c_{la}f_{B_{la}^{\eta}}}{K_{s}^{P4} + c_{la}} + Y_{pro}^{P4}k^{P4}f_{l} \frac{c_{la}f_{B_{la}^{\delta}}}{K_{s}^{P4} + c_{la}} $$

$$ G_{but} = Y_{but}^{P3}k^{P3}f_{l} \frac{c_{mon}f_{B_{mon}^{\phi}}}{K_{s}^{P3} + c_{mon}} + Y_{but}^{P3}k^{P3}f_{l} \frac{c_{mon}f_{B_{mon}^{\eta}}}{K_{s}^{P3} + c_{mon}} + Y_{but}^{P4}k^{P4}f_{l} \frac{c_{la}f_{B_{la}^{\phi}}}{K_{s}^{P4} + c_{la}} + \\ + Y_{but}^{P4}k^{P4}f_{l} \frac{c_{la}f_{B_{la}^{\eta}}}{K_{s}^{P4} + c_{la}} $$

$$ G_{AMPs} = -Y_{AMPs}^{P14}k^{P14}c_{AMPs} $$

$$ G_{H_2S} = Y_{H_2S}^{P2m}k^{P2}f_{l} \frac{0.05f_{sm}f_{B_{H2S}^{\delta}}}{K_{f}^{P2}f_{B_{H2S}^{\delta}} + 0.05f_{sm}} + Y_{H_2S}^{P2prot}k^{P2}f_{l} \frac{f_{prot}f_{B_{H2S}^{\delta}}}{K_{f}^{P2}f_{B_{H2S}^{\delta}} + f_{prot}} + \\ - k^{P11}(c_{H_2S} - K_{H_2S}^{P11}RT[H_2S_g]_{\infty}) $$

$$ G_{CH_4} = Y_{CH_4}^{P6}k^{P6}f_{l} \frac{c_{H_2}f_{B_{H2m}^{\phi}}}{K_{s}^{P6} + c_{H_2}} I_{pH} - k^{P11}(c_{CH_4} - K_{CH_4}^{P11}RT[CH_{4,g}]_{\infty}) $$

$$ G_{CO_2} = Y_{CO_2}^{P2}k^{P2}f_{l} \frac{0.05f_{sm}f_{B_{H2S}^{\delta}}}{K_{f}^{P2}f_{B_{H2S}^{\delta}} + 0.05f_{sm}} + Y_{CO_2}^{P2}k^{P2}f_{l} \frac{f_{prot}f_{B_{H2S}^{\delta}}}{K_{f}^{P2}f_{B_{H2S}^{\delta}} + f_{prot}} + $$
$$+ Y_{CO_2}^{P3} k^{P3} f_l \frac{c_{mon} f_{B_{mon}^{\phi}}}{K_s^{P3} + c_{mon}} + Y_{CO_2}^{P3} k^{P3} f_l \frac{c_{mon} f_{B_{mon}^{\eta}}}{K_s^{P3} + c_{mon}} +$$

$$+ Y_{CO_2}^{P4} k^{P4} f_l \frac{c_{la} f_{B_{la}^{\phi}}}{K_s^{P4} + c_{la}} + Y_{CO_2}^{P4} k^{P4} f_l \frac{c_{la} f_{B_{la}^{\eta}}}{K_s^{P4} + c_{la}} +$$

$$+ Y_{CO_2}^{P4} k^{P4} f_l \frac{c_{la} f_{B_{la}^{\delta}}}{K_s^{P4} + c_{la}} + Y_{CO_2}^{P5} k^{P5} f_l \frac{c_{H_2} f_{B_{H2a}^{\phi}}}{K_s^{P5} + c_{H_2}} +$$

$$+ Y_{CO_2}^{P6} k^{P6} f_l \frac{c_{H_2} f_{B_{H2m}^{\phi}}}{K_s^{P6} + c_{H_2}} I_{pH} - k^{P11} (c_{CO_2} - K_{CO_2}^{P11} RT[CO_{2,g}]_{\infty})$$

$$G_{H_2} = Y_{H_2}^{P2} k^{P2} f_l \frac{0.05 f_{sm} f_{B_{H2S}^{\delta}}}{K_f^{P2} f_{B_{H2S}^{\delta}} + 0.05 f_{sm}} + Y_{H_2}^{P2} k^{P2} f_l \frac{f_{prot} f_{B_{H2S}^{\delta}}}{K_f^{P2} f_{B_{H2S}^{\delta}} + f_{prot}} +$$

$$+ Y_{H_2}^{P3} k^{P3} f_l \frac{c_{mon} f_{B_{mon}^{\phi}}}{K_s^{P3} + c_{mon}} + Y_{H_2}^{P3} k^{P3} f_l \frac{c_{mon} f_{B_{mon}^{\eta}}}{K_s^{P3} + c_{mon}} +$$

$$+ Y_{H_2}^{P4} k^{P4} f_l \frac{c_{la} f_{B_{la}^{\phi}}}{K_s^{P4} + c_{la}} + Y_{H_2}^{P4} k^{P4} f_l \frac{c_{la} f_{B_{la}^{\eta}}}{K_s^{P4} + c_{la}} + Y_{H_2}^{P4} k^{P4} f_l \frac{c_{la} f_{B_{la}^{\delta}}}{K_s^{P4} + c_{la}} +$$

$$- k^{P5} f_l \frac{c_{H_2} f_{B_{H2a}^{\phi}}}{K_s^{P5} + c_{H_2}} - k^{P6} f_l \frac{c_{H_2} f_{B_{H2m}^{\phi}}}{K_s^{P6} + c_{H_2}} I_{pH} +$$

$$- k^{P11} (c_{H_2} - K_{H_2}^{P11} RT[H_{2,g}]_{\infty})$$

$$G_{O_2} = Y_{O_2}^{P2} k^{P2} f_l \frac{0.05 f_{sm} f_{B_{H2S}^{\delta}}}{K_f^{P2} f_{B_{H2S}^{\delta}} + 0.05 f_{sm}} + Y_{O_2}^{P2} k^{P2} f_l \frac{f_{prot} f_{B_{H2S}^{\delta}}}{K_f^{P2} f_{B_{H2S}^{\delta}} + f_{prot}} +$$

$$+ Y_{O_2}^{P4} k^{P4} f_l \frac{c_{la} f_{B_{la}^{\delta}}}{K_s^{P4} + c_{la}} - Y_{O_2}^{P13} k^{P13} c_{O_2}$$
**SM5 Secretion and absorption across the mucosal wall** In this section we report the calculations carried out to determine the secretion and absorption coefficients of the substances exchanged through the mucosal wall. These values are employed in the numerical codes and are summarized in Table SM2.

**SM5.1 Mucus production** Daily mucus production in the proximal colon over the domain considered in (6):

$$g_{m,tot} = \frac{5}{3} \cdot 20 \ [\text{cm}^3 \cdot \text{day}^{-1}]$$

where mucins production is $\frac{5}{3}$ [g · day$^{-1}$] and mucins are 5% of the total mucus. Daily mucus production per unit surface area:

$$g_m = \frac{g_{m,tot}}{\Gamma_l^{prox}} = \frac{5 \cdot 20}{3} \frac{1}{5 \cdot \pi \cdot 21} = 0.10 \ [\text{cm} \cdot \text{day}^{-1}]$$

Mucus production per Goblet cell:

$$(\text{SM5.1}) \qquad q_m = \frac{g_m}{N_{gc} \cdot N_{crypt}} = 1.20 \cdot 10^{-8} \ [\text{cm} \cdot \text{day}^{-1} \cdot \text{cell}^{-1}]$$

where $N_{gc}$ is the average number of Goblet cells per crypt.

**SM5.2 Water pumping** Daily water pumping in the proximal colon over the domain considered in (6):

$$g_{l,tot} = 1.60 \ [\text{day}^{-1}]$$

Daily water pumping per surface unit area:

$$g_l = \frac{g_{l,tot}}{\Gamma_l^{prox}} = \frac{1.60}{5 \cdot \pi \cdot 21} = 4.85 \cdot 10^{-3} \ [\text{cm}^{-2} \cdot \text{day}^{-1}]$$

Water pumping per Enterocyte cell:

$$(\text{SM5.2}) \qquad q_l = \frac{g_l}{N_{ent} \cdot N_{crypt}} = 1.91 \cdot 10^{-10} \ [\text{cm}^{-2} \cdot \text{day}^{-1} \cdot \text{cell}^{-1}]$$

where $N_{ent}$ is the average number of Enterocyte cells per crypt.

**SM5.3 Lactate absorption** Daily lactate absorption in the proximal colon over the domain considered in (6):

$$\gamma_{la,tot} = \gamma_{la}^{m} + \gamma_{la}^{l} = 12.60 + 0.88 = 13.48 \ [\text{day}^{-1}]$$

Daily lactate absorption per surface unit area:

$$\tilde{\gamma}_{la} = \frac{\gamma_{la,tot}}{\Gamma_l^{prox}} = 4.09 \cdot 10^{-2} \ [\text{cm}^{-2} \cdot \text{day}^{-1}]$$

Lactate absorption per Enterocyte cell:

$$(\text{SM5.3}) \qquad q_{la} = \frac{\tilde{\gamma}_{la}}{N_{ent} \cdot N_{crypt}} = 1.61 \cdot 10^{-9} \ [\text{cm}^{-2} \cdot \text{day}^{-1} \cdot \text{cell}^{-1}]$$
**SM5.4 Acetate absorption** Daily acetate absorption in the proximal colon over the domain considered in (6):

$$ \gamma_{ac,tot} = \gamma_{ac}^{m} + \gamma_{ac}^{l} = 1.32 + 18.90 = 20.22 \ [\text{day}^{-1}] $$

Daily acetate absorption per surface unit area:

$$ \gamma_{ac} = \frac{\gamma_{ac,tot}}{\Gamma_{l}^{prox}} = 6.13 \cdot 10^{-2} \ [\text{cm}^{-2} \cdot \text{day}^{-1}] $$

Acetate absorption per Enterocyte cell:

$$ (\text{SM5.4}) \qquad q_{ac} = \frac{\gamma_{ac}}{N_{ent} \cdot N_{crypt}} = 2.41 \cdot 10^{-9} \ [\text{cm}^{-2} \cdot \text{day}^{-1} \cdot \text{cell}^{-1}] $$

**SM5.5 Propionate absorption** Daily propionate absorption in the proximal colon over the domain considered in (6):

$$ \gamma_{pro,tot} = \gamma_{pro}^{m} + \gamma_{pro}^{l} = 1.07 + 15.32 = 16.39 \ [\text{day}^{-1}] $$

Daily propionate absorption per surface unit area:

$$ \gamma_{pro} = \frac{\gamma_{pro,tot}}{\Gamma_{l}^{prox}} = 4.97 \cdot 10^{-2} \ [\text{cm}^{-2} \cdot \text{day}^{-1}] $$

Propionate absorption per Enterocyte cell:

$$ (\text{SM5.5}) \qquad q_{pro} = \frac{\gamma_{pro}}{N_{ent} \cdot N_{crypt}} = 1.96 \cdot 10^{-9} \ [\text{cm}^{-2} \cdot \text{day}^{-1} \cdot \text{cell}^{-1}] $$

**SM5.6 Butyrate absorption** Daily butyrate absorption in the proximal colon over the domain considered in (6):

$$ \gamma_{but,tot} = \gamma_{but}^{m} + \gamma_{but}^{l} = 0.90 + 12.88 = 13.78 \ [\text{day}^{-1}] $$

Daily butyrate absorption per surface unit area:

$$ \gamma_{but} = \frac{\gamma_{but,tot}}{\Gamma_{l}^{prox}} = 4.18 \cdot 10^{-2} \ [\text{cm}^{-2} \cdot \text{day}^{-1}] $$

Butyrate absorption per Enterocyte cell:

$$ (\text{SM5.6}) \qquad q_{but} = \frac{\gamma_{but}}{N_{ent} \cdot N_{crypt}} = 1.64 \cdot 10^{-9} \ [\text{cm}^{-2} \cdot \text{day}^{-1} \cdot \text{cell}^{-1}] $$
**SM6 Epithelial dynamics: cell migration and fate** While the previous sections detailed the luminal and mucosal metabolic environment, this section describes the dynamics of the intestinal epithelium. The epithelium is modeled as a population of five distinct cell types: stem cells (sc), deep crypt secretory cells (dcs), progenitor cells (pc), goblet cells (gc), and enterocytes (ent).

The spatial distribution and migration of cell type $\ell \in C = \{sc, dcs, pc, gc, ent\}$ are defined by their density $\rho_\ell(z, t)$ along the crypt's longitudinal axis $z \in [0, Z_{max}]$. Cell migration is governed by a system of nonlinear degenerate parabolic equations for all types except the stationary dcs:

$$ (\text{SM6.1}) \quad \left\{ \begin{array}{l} \partial_t \rho_\ell + W \partial_z (\phi(z) (\partial_z \bar{\rho} + \alpha \bar{\rho}) \rho_\ell) = \sum_k \eta_{k,\ell} q_k Q_k(\ell, z, \bar{\rho}, \tilde{c}_{but}, f_B), \\ \rho_\ell(0, t) = \rho_\ell^{bot}, \\ \partial_z \rho_\ell(Z_{max}, t) = 0. \end{array} \right. $$

Here, $\bar{\rho}(z, t) = \sum_{\ell \in C} \rho_\ell(z, t)$ represents the total local cell density. The coefficient $\phi(z)$ accounts for the curvature effects inherent to the crypt geometry, ensuring a consistent representation of the interaction forces within the truncated-tube structure:

$$ (\text{SM6.2}) \quad \phi(z) = \begin{cases} \frac{\bar{f}(z)}{r_0} - \frac{\bar{f}(0)}{r_0} & \text{if } z \le r_0 - \varepsilon, \\ 1 & \text{if } r_0 - \varepsilon < z < Z_{max} - r_0 + \varepsilon, \\ \frac{\bar{f}(Z_{max}) - \bar{f}(z)}{r_0} - \frac{\bar{f}(z)}{r_0} & \text{if } Z_{max} - r_0 + \varepsilon \le z, \end{cases} $$

where $\tilde{f}(z)$ defines the crypt radius along the axis:

$$ (\text{SM6.3}) \quad \tilde{f}(z) = \begin{cases} r_0 \sqrt{\frac{z+\varepsilon}{r_0} \left(2 - \frac{z+\varepsilon}{r_0}\right)} & \text{if } z \le r_0 - \varepsilon, \\ r_0 & \text{if } r_0 - \varepsilon < z < Z_{max} - r_0 + \varepsilon, \\ r_0 \left(2 - \sqrt{\frac{Z_{max}-z+\varepsilon}{r_0} \left(2 - \frac{Z_{max}-z+\varepsilon}{r_0}\right)}\right) & \text{if } Z_{max} - r_0 + \varepsilon \le z. \end{cases} $$

The source term in Eq. (SM6.1) represents cell fate events, where $\eta_{k,\ell}$ identifies whether an event $k$ produces or consumes cell type $\ell$:

$$ (\text{SM6.4}) \quad \eta_{k,\ell} = \begin{cases} \rho_\ell & \text{if } k \in \text{Div}, \\ -\rho_\ell & \text{if } k \in \text{Ex}, \\ \rho_{\ell_k} & \text{if } k \in \text{Dif} \text{ (differentiation from } \ell_k \text{ to } \ell), \\ -\rho_\ell & \text{if } k \in \text{Dif} \text{ (differentiation from } \ell \text{).} \end{cases} $$

We distinguish between division events ($\text{Div} = \{\text{div}, \text{sc}; \text{div}, \text{pc}\}$), extrusion events ($\text{Ex} = \{\text{ex}, \text{gc}; \text{ex}, \text{ent}\}$), and differentiation paths ($\text{Dif} = \{\text{sc}, \text{pc}; \text{pc}, \text{gc}; \text{pc}, \text{ent}\}$). The maximal rate of cell fate event is given by the positive constant $q_k$ and the function $Q_k(\cdot)$ is linked to regulation mechanisms. Function $Q_k$ is a combination of
the activation function $R(y, K, \kappa)$ defined as:

(SM6.5)

$$R(y, K, \kappa) := \begin{cases} 0 & \text{if } y \le K - \kappa, \\ -\frac{y^3}{4\kappa^3} + \frac{3K}{4\kappa^3} - \frac{(3K^2 - 3\kappa^2)y}{4\kappa^3} + \frac{K^3 + 2\kappa^3 - 3K\kappa^2}{4\kappa^3} & \text{if } K - \kappa < y < K + \kappa, \\ 1 & \text{if } K + \kappa \le y. \end{cases}$$

for $K > 0$ and $\kappa > 0$, and its detailed form is listed below.

For stem cells:

(SM6.6)

$$Q_{\text{div,sc}}(\ell, z, \bar{\rho}(z), \tilde{c}_{\text{but}}(z), f_B) = \mathbf{1}_{\ell=\text{sc}}(1 - R(z, K_{\text{div,sc}}[z], \kappa_{\text{div,sc}}[z])) \\ \times (1 - R(D\rho(z, t), K_{\text{div,sc}}[\text{dens}], \kappa_{\text{div,sc}}[\text{dens}])) \\ \times (1 - R(\tilde{c}_{\text{but}}(z), K_{\text{div,sc}}[\text{but}], \kappa_{\text{div,sc}}[\text{but}])) \\ \times (R(f_B, K_{\text{div,sc}}[\text{PRRs}], \kappa_{\text{div,sc}}[\text{PRRs}]).$$

$$Q_{\text{sc,pc}}(\ell, z) = \mathbf{1}_{\ell=\text{sc}} R(z, K_{\text{sc,pc}}[z], \kappa_{\text{sc,pc}}[z]).$$

For progenitor cells:

$$Q_{\text{div,pc}}(\ell, z, \bar{\rho}(z), f_B) = \mathbf{1}_{\ell=\text{pc}}(1 - R(z, K_{\text{div,pc}}[z], \kappa_{\text{div,pc}}[z])) \\ \times (1 - R(D\rho(z), K_{\text{div,pc}}[\text{dens}], \kappa_{\text{div,pc}}[\text{dens}])) \\ \times (R(f_B, K_{\text{div,pc}}[\text{PRRs}], \kappa_{\text{div,pc}}[\text{PRRs}]).$$

$$(\text{SM6.7}) \qquad Q_{\text{pc,gc}}(\ell, z, \tilde{c}_{\text{but}}(z)) = \mathbf{1}_{\ell=\text{pc}} R(z, K_{\text{pc,gc}}[z], \kappa_{\text{pc,gc}}[z]) \\ \qquad \qquad \times R(\tilde{c}_{\text{but}}(z), K_{\text{pc,gc}}[\text{but}], \kappa_{\text{pc,gc}}[\text{but}]).$$

$$Q_{\text{pc,ent}}(\ell, z, \tilde{c}_{\text{but}}(z)) = \mathbf{1}_{\ell=\text{pc}} R(z, K_{\text{pc,ent}}[z], \kappa_{\text{pc,ent}}[z]) \\ \times R(\tilde{c}_{\text{but}}(z), K_{\text{pc,ent}}[\text{but}], \kappa_{\text{pc,ent}}[\text{but}]).$$

For goblet cells and enterocytes:

$$Q_{\text{ex,gc}}(\ell, z, \bar{\rho}(z)) = \mathbf{1}_{\ell=\text{gc}} R(z, K_{\text{ex,gc}}[z], \kappa_{\text{ex,gc}}[z]) \\ \times R(D\rho(z), K_{\text{ex,gc}}[\text{dens}], \kappa_{\text{ex,gc}}[\text{dens}]).$$

(SM6.8)

$$Q_{\text{ex,ent}}(\ell, z, \bar{\rho}(z)) = \mathbf{1}_{\ell=\text{ent}} R(z, K_{\text{ex,ent}}[z], \kappa_{\text{ex,ent}}[z]) \\ \times R(D\rho(z), K_{\text{ex,ent}}[\text{dens}], \kappa_{\text{ex,ent}}[\text{dens}]).$$

Following (2), we assume DCS cells are stationary and do not undergo division or differentiation. Although they are biologically inactive in the model, they exert a critical mechanical force on neighboring cells due to their volume-occupying presence at the crypt base. This physical presence influences the total density gradient $\partial_z \rho$ in Eq. (SM6.1). The DCS density profile is described by:

$$(\text{SM6.9}) \qquad \rho_{\text{dc}s}(z) = \frac{N_{\text{dc}s}}{\int_{0}^{z_{\text{max}}} \mathcal{D}(z) dz} \cdot \mathcal{D}(z),$$

where the distribution function $\mathcal{D}(z)$ is given by:

$$(\text{SM6.10}) \qquad \mathcal{D}(z) = 1 - \mathcal{R}(z; K, \kappa),$$

and the regulation function $\mathcal{R}(z; K, \kappa)$, with parameters $K = \kappa = 12$, is defined as in Eq. (SM6.5).
**SM7 Metabolites and oxygen reaction-diffusion** In the crypt, SCFAs generated by microbial metabolism diffuse until they are either absorbed or metabo-lized by the epithelium. Butyrate plays a crucial role in the host-microbiota symbiosis, as its presence is associated with the activation of β-oxidation, a metabolic process in epithelial cells consuming O₂.

The concentrations of metabolites $\tilde{c}_i(z,t)$ $i \in \mathcal{M} = \{la, ac, pro, but\}$ in the crypt evolve according to the following reaction-diffusion equation:

$$ (\text{SM7.1}) \quad \left\{ \begin{array}{l} \partial_t \tilde{c}_i - \sigma_i \partial_{zz} \tilde{c}_i = -s_i \beta(\tilde{c}_i, \tilde{c}_o, \tilde{c}_{but}, c_{H_2S}(t), \rho_{gc}(z,t) + \rho_{ent}(z,t)) - \gamma_i^\alpha \rho_{ent}(z,t) \tilde{c}_i, \\ \tilde{c}_i(Z_{max}, t) = c_i(t), \\ \partial_z \tilde{c}_i(0, t) = 0. \end{array} \right. $$

For each metabolite, the diffusion rate is denoted by $\sigma_i$, the absorption depends on enterocytes density $\rho_{ent}$ and occurs at rate $\gamma_i^\alpha$. The equivalent metabolite concentra-tion obtained in the colon model $c_i(t)$ is found in the Dirichlet boundary condition at the top of the crypt. β-oxidation is preceded by a stoichiometric coefficient $s_i$ and is given by :

$$ (\text{SM7.2}) \quad \beta(\tilde{c}_i, \tilde{c}_o, \tilde{c}_{but}, c_{H_2S}^L, \rho_{gc} + \rho_{ent}) = \gamma_i^\beta \frac{\tilde{c}_o^{s_{o,i}} \tilde{c}_i^{s_i}}{\tilde{c}_o^{s_{o,i}} \tilde{c}_i^{s_i} + K_i^{s_{o,i}+s_i}} (\rho_{gc} + \rho_{ent}) \\ \times R(\tilde{c}_{but}(z,t), K_{HIF}, \kappa_{HIF}) \\ \times (1 - R(c_{H_2S}(t), K_{H_2S}, \kappa_{H_2S})) $$

where $\tilde{c}_o$ is the oxygen concentration and $s_{o,i}$ is the stoichiometric coefficient of oxygen in the reaction. The maximum reaction rate is represented by $\gamma_i^\beta$, the affinity constant of β-oxidation is denoted by $K_i$, the density of differentiated cells in the crypt is calculated as $(\rho_{gc} + \rho_{ent})$. The regulatory pathways of β-oxidation are governed by the regulatory function $R(\tilde{c}_{but}, K_{HIF}, \kappa_{HIF})$ and $(1 - R(c_{H_2S}, K_{H_2S}, \kappa_{H_2S}))$ from Eq. SM6.5. Additionally, $c_{H_2S}(t)$ is the concentration of $H_2S$ in the lumen of the colon model.

Oxygen, supplied by the blood vessels in the colon wall, diffuses into the crypt and is consumed by enterocytes and goblet cells during β-oxidation, giving the diffusion-reaction PDE :

$$ (\text{SM7.3}) \quad \left\{ \begin{array}{l} \partial_t \tilde{c}_o - \sigma_o \partial_{zz} \tilde{c}_o = - \sum_{i \in \mathcal{M}} s_{o,i} \beta(\tilde{c}_i, \tilde{c}_o, \tilde{c}_{but}, c_{H_2S}, \rho_{gc} + \rho_{ent}), \\ \tilde{c}_o(0, t) = \tilde{c}_o^{\text{bot}}, \\ \partial_z \tilde{c}_o(Z_{max}, t) = 0. \end{array} \right. $$
## SM8 Additional simulation results

<figure><img src="image_17.png" /></figure>

Fig. SM2: Soft mucus distribution (left panel) and cross-sectional profile at mid-length of the domain (right panel) under a health reference diet. The dashed line corresponds to the isoline at a soft mucus volume fraction of $f_{sm} = 0.05$, which is taken in our model as the threshold defining the soft mucus layer thickness.

<figure><img src="image_18.png" /></figure>

Fig. SM3: Hard mucus distribution (left panel) and cross-sectional profile at mid-length of the domain (right panel) under a health reference diet. The dashed line corresponds to the isoline at a hard mucus volume fraction of $f_{hm} = 0.01$, which is taken in our model as the threshold defining the thickness of the hard mucus layer.
<figure><img src="image_19.png" /><figcaption>Fig. SM4: Spatial distribution of the total bacterial population within the colon domain (left panel) and radial profile obtained from a cross-section taken at mid height of the computational domain (right panel).</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. SM5: Composite view of O₂ dynamics across the three colon segments. Each panel presents the time evolution of the O₂ concentration at the crypt top for each colonic segment.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. SM6: (a) Cell densities and (b) normalized solutes concentration along the Average Crypt 1 within the O₂ bloom scenario.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Fig. SM7: (a) **Cell densities** and (b) **normalized solutes concentration** along the Average Crypt 2 within the O₂ bloom scenario.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Fig. SM8: (a) **Cell densities** and (b) **normalized solutes concentration** along the Average Crypt 3 within the O₂ bloom scenario.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. SM9: **Cross-sectional profiles of hard mucus volume fraction across the three average crypts under the reference diet.** In each figure, the impaired scenario (solid line) is compared against the healthy baseline case (dotted line).</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Fig. SM10: **AMPs concentration profiles: baseline vs. impaired scenario.**
Concentration of Antimicrobial Peptides (AMPs) plotted across the radial cross-sections in each section. The dashed lines represent baseline healthy conditions, and the solid lines represent the impaired scenario.</figcaption></figure>

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