Model

Introduction

This study investigates the physicochemical properties of protein adhesions under vacuum conditions to model their performance in repairing spaces suit breaches s. By applying physical property tests and mathematical modeling, we examined protein propagation, locking time, bridge formation time as well as physical properties in vacuum. The model aims s to ensure both rapid repair capability and long-term stability of the material.

Physical Property Testing Model of Repaired Protein

Assumptions and Parameters

Symbol	Definition
A / m²	Total adhesion area
r / m	Radius of spacesuit breach
R / m	Radius of gel layer formed by overflow
L / m	Penetration depth into the spacesuit
N / N/m²	Protein adhesion per unit area
F / N	Total force applied
P / Pa	Internal pressure of the spaces suit
V / m³	Total volume of protein used
H / m	Thickness of the upper gel layer

Basic Assumptions s

1. The breach in the spacesuit is assumed to be circular, calculable by A=πr². Similarly, the gel layer formed by protein overflow is also circular.
2. Deformation of the gel layer due to pressure difference is negligible.
3. The protein mixture consists of Alfalfa5 and Gamma5. Relevant data can be found in the references.

Model Used

Based on the mechanical balance and strength assessment model of a thin-walled cylindrical shell:

• The cylinder is thin-walled, homogeneous, and isotropic.
• Only uniform internal pressure P is considered; axial bending and local stress concentration are neglected.
• Failure criterion: Total internal force ≥ Total internal pressure force, i.e.,

Static Stress Analysis of the Protein Gel Layer

Based on the model, the following derivation and 2D visualization were performed:

Conclusion

Through static stress analysis based on the thin-walled cylindrical shell model, it is concluded that the protein gel layer effectively withstands internal pressure when there exists an equilibrium between static stress and the pressure difference across the spacesuit. Verified via 2D visualization, the relationship between the gel radius R and the breach radius r satisfies the following strength condition:

where A is the load-bearing cross-sectional area, N is the adhesive force per unit area, and F is the total axial force due to internal pressure.

Mathematical Relationship Between Protein Volume Required and Repair Size

Based on the previous static stress analysis and the simulated protein gel layer image, the corresponding volume can be calculated.

No additional assumptions.

Model Used

Axial force-area strength assessment model.

Key Equations

Derivation Based on the Model

The volume calculation is derived from the geometric parameters of the gel layer and breach, combining the load-bearing area and penetration depth to ensure sufficient material for repair.

Conclusion

Using the axial force-area strength assessment model, the mathematical relationship between the protein volume V required for repair and the breach area A is derived. Under constant protein concentration, the volume needed to repair breaches of different sizes satisfies:

This relationship indicates that the protein volume is positively correlated with the breach radius r and the gel extension radius R, providing a theoretical basis for determining protein usage in practical repairs.

Mathematical Relationship Between Protein Repair Time and Strength of Formed Protein Bridges

The forementioned two models suggest that in space, protein released from capsules does not instantly form bridges that are capable of withstanding the pressure difference. Therefore, a mathematical model for protein bridge formation time is developed.

Additional Parameters

Symbol	Definition
D₀ / m²/s	Diffusion coefficient of protein in water at 1 atm [1]
Dvac / m²/s	Effective diffusion coefficient in vacuum
Kn	Knudsen number (mean free path to characteristic length)
λ / m	Molecular mean free path
R / m	Spacesuit thickness
Dsurf / m²/s	Diffusion coefficient on the spacesuit surface [3]
t(diff) / s	Diffusion time required
k₁' / s⁻¹	Effect of vacuum and low temp. on protein binding [2]
B(t) / N	Number of protein bridges formed at time t
B(max) / N	Number of bridges required for original strength
t₁ / s	Time from capsule burst to protein arrival at breach
k₂' / s⁻¹	Effect of vacuum and low temp. on protein locking [2]
t (total) / s	Total time required
t (max) / s	Maximum protein locking time
[P] / µm⁻²	Local concentration after release

Additional Assumptions

1. Load distribution assumption: Bridges share the load uniformly.
2. Bridge independence assumption: Each protein bridge functions independently.
3. Material strength and uniformity: Interface strength is identical and constant across all bridges.
4. Static environment assumption: Vacuum and low temperature affect binding via k₁' and locking via k₂'.
5. Time scale separation: Different protein stages are temporally independent.
6. Molecular mean free path λ is set to 65 nm for calculating Dvac
7. Capsules burst at a distance of 1 µm from the breach.

Reference Models

1. 1. Diffusion Model



Effective diffusion coefficient:

Dvac = D0/μ × Kn



Knudsen number:

Kn = μ/R



Diffusion time:

t(diff) = L²/(4Ddiff)



2. 2. Adsorption/Repair Kinetics Model (Exponential Growth)



Protein bridges formed over time:

B(t) = B(max) · [1 – exp(–k(t) · t)]



Time required to achieve repair:

t = –ln(1 – B(t)/B(max)) / k(t)

Derivation Based on Models and Assumptions

Conclusion

Model calculations indicate that while protein diffusion rate is significantly enhanced in vacuum, its anchoring and locking rates are regulated by environmental factors k₁' and k₂'. The total repair time t(total) is the sum of diffusion time, anchoring time, and protein bridge growth time. This time establishes a clear mathematical relationship with the final number of protein bridges B(t), i.e., the repair strength, providing a theoretical basis for predicting in-orbit repair efficacy.

References

Some experimental data in Models 1 and 2 are derived from our wet-lab experiments.

1. Yarger, J. L., Cherry, B. R., & van der Vaart, A. (2018). Uncovering the structure–function relationship in spider silk. Nature Reviews Materials. https://doi.org/10.1038/natrevmats.2018.8
2. Vailati, A., Šeta, B., Bou-Ali, M. M., & Shevtsova, V. (2024). Perspective of research on diffusion: From microgravity to space exploration. International Journal of Heat and Mass Transfer, 222, 125705. https://doi.org/10.1016/j.ijheatmasstransfer.2024.125705
3. Kuang, Z. (2018, June 27). Multiscale modeling and simulation of protein adsorption on surfaces [Lecture presentation]. School of Mathematical Sciences, Xiamen University, Xiamen, China. https://math.xmu.edu.cn/info/1034/9347.htm