Model

Estimation of GZMK Concentration in Mucus

Background

Based on the dual role of Granzyme K (GZMK) as both a diagnostic biomarker and a therapeutic target in Chronic Rhinosinusitis (CRS), we initiated the DOCTOR project to explore new strategies for the early diagnosis and precision treatment of CRS. For the diagnostic aspect, we developed a colloidal gold test strip designed to detect GZMK in nasal mucus. However, existing research does not directly confirm whether GZMK is present in nasal (sinus) mucus. To estimate the approximate concentration of the GZMK protein in the nasal (sinus) mucus, we established a diffusion-equilibrium mathematical model. This model simulates two core biological processes: GZMK diffusing from inflamed nasal tissue into the mucus, and GZMK being cleared as the mucus is renewed (via ciliary movement). By balancing these two processes, we calculated the steady-state concentration of GZMK in the mucus, which provides a theoretical basis for detection with the colloidal gold test strip.

Assumptions

1. Constant GZMK concentration in tissues

   In the state of rhinosinusitis, the rates of GZMK production and consumption in the sinus mucosal tissue are relatively stable. Thus, the tissue concentration can be considered constant over a certain period (several hours to 1 day, which is much longer than the recover cycle).

2. Dynamic equilibrium of the system

   Mucus is continuously renewed through cilia swing to push mucus toward the nasopharynx for excretion. After GZMK enters mucus from tissues, it is continuously carried away with mucus renewal. When the transport rate of GZMK from tissues to mucus equals the clearance rate of GZMK in mucus, the concentration in mucus no longer changes, i.e., a steady state is reached.

Model Establishment

According to Fick's first law, the diffusion rate of GZMK from tissues to mucus is expressed as:

$$ J_{\text{diff}} = P \cdot A \cdot (C_t - C_m) $$

where $ P $ is the permeability coefficient, describing the difficulty of GZMK crossing the nasal polyp mucosal barrier; $ A $ is the diffusion area, i.e., the anatomical area of the polyp; $ C_t $ and $ C_m $ are the GZMK concentrations in tissues and mucus, respectively.

The clearance rate of GZMK in mucus is determined by mucus renewal. At steady state, the total amount of mucus remains constant, and the amount of mucus cleared equals the amount produced. Thus, the clearance rate of GZMK is:

$$ J_{\text{clear}} = Q \cdot C_m $$

where $ Q $ is the volume of mucus excreted from the sinus per unit time. At equilibrium, the total amount of GZMK diffusing from tissues into mucus is exactly equal to the total amount cleared by mucus renewal, i.e.:

$$ P \cdot A \cdot (C_t - C_m) = Q \cdot C_m $$

Rearranging the equation gives:

$$ C_m = \frac{P \cdot A}{P \cdot A + Q} \cdot C_t $$

In rhinosinusitis, the permeability coefficient $ P $ of GZMK across nasal polyp mucosa is essentially determined by both the degree of membrane damage due to inflammation and the molecular properties of GZMK itself, which can be expressed as:

$$ P \approx K \cdot P_0 $$

where $ P_0 $ is the baseline permeability coefficient of normal nasal mucosa; $ K $ is the barrier damage coefficient of rhinosinusitis. For different proteins, their permeability is inversely proportional to the square root of their molecular weight:

$$ P_0 \propto \frac{1}{\sqrt{MW}} $$

In rhinosinusitis, the tight junctions of nasal mucosal epithelium are damaged, leading to impaired barrier function, which is manifested by a decrease in transepithelial electrical resistance (TEER). Thus, the barrier damagrhinosinusitissitis can be quantified by TEER:

$$ K = \frac{P}{P_0} = \frac{TEER_0}{TEER} $$

where $ TEER_0 $ and $ TEER $ are the TEER values of normal and diseased nasal mucosa, respectively.

Parameter Determination

Currently, population-level data on the surface area of nasal polyps is lacking. To establish a representative order of magnitude, we referred to the dimensions of medium-to-large polyp specimens reported in the literature and approximated the surface area using the formula for an ellipsoid, yielding an estimate of $ 33–41 \text{cm}^2 $. Consequently, we have adopted $ 35 \text{cm}^2 $ as a typical surface area value for moderate-to-severe chronic rhinosinusitis with nasal polyps in our subsequent mode[3].

The mucus renewal rate $ Q $ is approximately $ 1.8 \text{mL/h} = 5 \times 10^{-4} \text{mL/s} $[4].

Lang, S. et al. measured the apparent permeability coefficients $P$ of somatostatin analog octreotide (SMS) and thymopeptide fragment TP4 through ex vivo bovine nasal mucosa[1]. The part of their results are summarized in the table below, along with their molecular weights (MW) and the calculated values of $P \cdot \sqrt{MW} $:

Substance	Molecular Weight (MW, g/mol)	Apparent Permeability Coefficient (P, cm/s)	$ P \cdot \sqrt{MW} $
Octreotide (SMS)	1079.3	$ 4 \times 10^{-5} $	0.0013
Thymopeptide fragment (TP4)	4963.4	$ 1.7 \times 10^{-5} $	0.0012

The close values of $ P \cdot \sqrt{MW} $ for SMS and TP4 (0.0013 and 0.0012, respectively) confirm the relationship $ P \propto \frac{1}{\sqrt{MW}} $, as this product remains approximately constant for different substances when the proportionality holds. Based on this relationship, the baseline permeability coefficient $ P_0 $ for GZMK (with a molecular weight of 32 kDa) is calculated to be in the range of $ [6.695 \times 10^{-4}, 7.346 \times 10^{-4}] \text{cm/s} $, which matches the consensus that for proteins with a molecular weight of ~30 kDa (e.g. GZMK), the permeability coefficient through nasal mucosa is reported to be in the range of $10^{-4} \sim 10^{-3} \text{cm/h}$ .

The TEER of in vitro primary human nasal mucosal epithelial monolayers is approximately $ 300 \sim 500 \Omega \cdot \text{cm}^2 $ (most literature uses the median value of $ 400 \Omega \cdot \text{cm}^2 $); in acute bacterial rhinosinusitis, TEER decreases to $ 120 \sim 200 \Omega \cdot \text{cm}^2 $; in chronic rhinosinusitis, TEER further decreases to $ 80 \sim 150 \Omega \cdot \text{cm}^2 $. Thus, the barrier damage coefficient $ K $ is in the range of $ [2, 5] $.

Accordingly, the permeability coefficient $ P $ of GZMK is calculated to be in the range of $ [1.339 \times 10^{-5}, 3.673 \times 10^{-5}] \text{cm/s} $.

The GZMK concentration in inflamed nasal polyp tissues is $ C_t = 6 \text{ng/mL} $.

Substituting the parameters into the equation for $ C_m $:

$$ C_m = \frac{P \cdot A}{P \cdot A + Q} \cdot C_t \in [8.03 \times 10^{-5}, 22.04 \times 10^{-5}] \text{ng/mL} $$

Reference

[1] Lang, Steffen, Barbara Rothen-Rutishauser, Jean-Claude Perriard, M. Christiane Schmidt, and Hans P Merkle. ‘Permeation and Pathways of Human Calcitonin (hCT) Across Excised Bovine Nasal Mucosa’. Peptides 19, no. 3 (1998): 599–607. https://doi.org/10.1016/S0196-9781(97)00470-1.

[2] Ramezanpour, Mahnaz, Sholeh Feizi, Hashan Dilendra Paththini Arachchige, et al. ‘Evaluation of Mucosal Barrier Disruption Due to Staphylococcus Lugdunensis and Staphylococcus Epidermidis Exoproteins in Patients with Chronic Rhinosinusitis’. International Forum of Allergy & Rhinology 15, no. 3 (2025): 267–77. https://doi.org/10.1002/alr.23481.

[3] Testa D, Nunziata M, Romano ML, Massimilla EA, Toni G, De Cristofaro G, Marcuccio G, Motta G. Choanal Polyp with Osseous Metaplasia: Radiological and Therapeutic Management of a Rare Case and a Review of Bone Metaplastic Lesions of Sinonasal Tract. Am J Case Rep. 2020 Mar 30;21:e921494. doi: 10.12659/AJCR.921494.

[4]Gizurarson S. The effect of cilia and the mucociliary clearance on successful drug delivery. Biol Pharm Bull. 2015;38(4):497-506. doi: 10.1248/bpb.b14-00398. Epub 2015 Feb 7. PMID: 25739664.

[5]Lan, F., Li, J., Miao, W. et al. GZMK-expressing CD8+ T cells promote recurrent airway inflammatory diseases. Nature 638, 490–498 (2025). https://doi.org/10.1038/s41586-024-08395-9

GZMK Active Structure Prediction and Refinement

Since accurate structural information is indispensable for inhibitor design and virtual screening, our first step was to reconstruct the active conformation of GZMK. The only crystal structure currently available is a catalytically inactive S195A pro-enzyme mutant, which was engineered to avoid autocleavage but does not represent the native functional state. To overcome this limitation, we employed a combination of AlphaFold-based prediction and molecular dynamics (MD) refinement to obtain a reliable structural model under near-physiological conditions.

We began by submitting the full-length GZMK amino acid sequence to the AlphaFold server, which generated five candidate models. The best-ranked structure achieved a predicted TM-score:

$$ \text{pTM} \approx 0.92 $$

indicating high confidence in its global fold.

Structural alignment against the pro-enzyme crystal structure confirmed that the predicted fold retained the expected serine protease backbone architecture. Alignment quality was quantified by the root-mean-square deviation (RMSD) between Cα atoms:

$$ \text{RMSD} = \sqrt{\frac{1}{N}\sum_{i=1}^N \left| \mathbf{r}_i^{\text{pred}} - \mathbf{r}_i^{\text{exp}} \right|^2} $$

where $N$ is the number of aligned atoms, $\mathbf{r}_i^{\text{pred}}$ are coordinates from the AlphaFold model, and $\mathbf{r}_i^{\text{exp}}$ are from the crystal structure. RMSD values below $\sim 2$ Å indicated strong topological consistency. This phase established a high-fidelity initial model.

A potential concern was whether post-translational modifications, particularly N-glycosylation, might alter the protein’s surface chemistry and confound hotspot selection. To evaluate this, we applied the Re-Glyco tool from the GlycoShape platform, which computationally screens for canonical N–X–S/T motifs with sufficient solvent accessibility. No reliable glycosylation sites were detected on the mature protein, suggesting that surface interaction pockets would not be significantly affected by glycan modifications. This outcome allowed us to focus our subsequent efforts on structural and energetic hotspot identification, free from the uncertainty of unknown PTMs.

While AlphaFold provides reliable backbone topology, it is less accurate for side-chain orientations and flexible loops, which are essential for ligand docking. To refine the model, we performed a 100 ns MD simulation using the Desmond module under near-physiological conditions (310 K, 0.15 M NaCl, S-OPLS force field). The time evolution of the structure was monitored by RMSD($t$):

$$ \text{RMSD}(t) = \sqrt{\frac{1}{N}\sum_{i=1}^N \left| \mathbf{r}_i(t) - \mathbf{r}_i(0) \right|^2} $$

where $\mathbf{r}_i(t)$ is the position of atom $i$ at time $t$, and $\mathbf{r}_i(0)$ is its reference position.

Throughout the trajectory, RMSD fluctuations remained below 3.8 Å, demonstrating the global stability of the protein structure and the absence of large-scale structural drift. Conformational clustering identified a dominant equilibrium state, and the centroid of this cluster was selected as the final optimized model.

Through this integrative approach—AlphaFold prediction, glycosylation site screening, and MD relaxation—we obtained a structurally and dynamically stable model of wild-type GZMK. This optimized structure provides not only a robust framework for defining druggable pockets but also a realistic basis for subsequent energetic analyses and inhibitor screening.