Hybrid Kinetic Modelling of Protein Crystallization: Hanging Drop and Langmuir–Blodgett Conditions
Abstract
1. Introduction
2. Methods
2.1. Protein Selection and Kinetic Data Processing
2.2. Modeling Crystallization Kinetics: Classical, Cooperative, and Diffusion–Limited Regimes
2.3. Model Fitting and Descriptor Extraction
- (i)
- crystallization half–time (t1⁄2, when X(t) reaches 50% of its plateau), which describes the timing of crystallization onset.
- (ii)
- time of maximum growth rate (tmax) which identifies the point of maximum growth acceleration.
- (iii)
- the peak growth rate (dX/dt)max which measures the intensity of the crystallization process.
- (iv)
- width at half–maximum (W1⁄2) of the dX/dt profile quantifying the duration of the cooperative growth phase.
2.4. Statistical Evaluation of Model Fits
3. Results: Growth and Rate Profiles
3.1. Lysozyme—HD and LB Kinetics

3.2. Thaumatin: Accelerated and Amplified Growth Under LB Conditions

3.3. Ribonuclease A: Sharp Transition and High Synchrony Under LB Conditions

3.4. Proteinase K: Diffuse Growth with Weak LB Response and Limited Experimental Resolution

3.5. Overall Results
3.6. Statistical Evaluation of Model Fits
4. Discussion: Interpretation of Kinetic Descriptors
4.1. Comparative Interpretation of Crystallization Profiles
4.2. Descriptor-Based Kinetic Analysis and Model Performance



4.3. Descriptor Ratios: Cross-System Mechanistic Signatures
4.4. Model Families, Descriptor Robustness, and Cross-System Interpretation
- Lysozyme: All models capture LB-induced shifts to earlier t1/2 and tmax. Mechanistic fits produce higher peaks and narrower W1/2 while sigmoid models yield smoother, broader transitions. This suggests LB promotes earlier onset with only moderate synchronization, consistent with a modest narrowing of W1⁄2.
- Thaumatin: LB consistently increases growth rates and advances tmax, yet W1/2 remains wide, but (dX/dt)max rises strongly, indicating intensified but still temporally extended growth. This indicates acceleration without sharp temporal confinement, and model outputs agree closely.
- Ribonuclease A: Model choice influences interpretation most strongly. Logistic and Hill fits both show advanced onset and steeper slopes, with Hill notably indicating tmax < t1/2, consistent with synchronized bursts. The Logistic model underestimates this burst, while Hill more faithfully captures the cooperative peak. Avrami and Kashchiev vary more in peak width but confirm accelerated crystallization.
- Proteinase K: Sparse data preclude reliable conclusions. Logistic and GSM yield interpretable trends; Avrami and Kashchiev produce unstable or nonphysical parameters (e.g., inflated peak rates, near-zero W1/2). Hill over-concentrates the profile. Only shape-controlled models provide reliable fits.
4.5. Model Discrimination via AIC Analysis
5. Conclusions
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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| Protein | Model | Best Fit Parameters HD | Best Fit Parameters LB |
|---|---|---|---|
| Lysozyme | Avrami | k = 5.30, n= 0.73, τ = 1.73 | k = 1.54, n = 0.732, τ = 0.798 |
| Kashchiev | k = 1.54, n = 2.96, τ = 1.178 | k = 3.426, n= 1.12, τ = 0.7979 | |
| Hill | k = 1.684, n= 6.02 | k = 1.14, n= 2.72 | |
| Logistic | k = 5.841, t0 = 1.768 | k = 2.049, t0 = 1.197 | |
| GSM | k = 7.289, t0 = 1.768, = 1.51 | k = 2.047, t0 = 1.197, m = 0.979 | |
| Thaumatin | Avrami | k = 0.5186, n= 1, τ = 2.784 | k = 0.6467, n= 1, τ = 2.552 |
| Kashchiev | k = 0.9041, n= 1.24, τ = 2.784 | k = 1.181, n = 1.26, τ = 2.552 | |
| Hill | k = 3.977, n = 3.5 | k = 3.646, n= 3.25 | |
| Logistic | k = 0.8501, t0 = 4.176 | k = 0.7873, t0 = 3.829 | |
| GSM | k = 0.823, t0 = 4.176, m = 0.925 | k = 0.8236, t0 = 3.829, m = 1.16 | |
| Ribonuclease A | Avrami | k = 0.8007, n= 2.65, τ = 1.075 | k = 1.66, n = 1, τ = 0.566 |
| Kashchiev | k = 1.211, n = 5.88, τ = 1.205 | k = 3.32, n= 1.13, τ = 0.566 | |
| Hill | k = 0.9369, n= 1.49 | k = 0.8086, n= 1.83 | |
| Logistic | k = 1.267, t0 = 0.9837 | k = 2.305, t0 = 0.849 | |
| GSM | k = 1.207, t0 = 0.9837, m = 0.679 | k = 1.843, t0 = 0.849, m = 0.799 | |
| Proteinase K | Avrami | k = 48.77, n= 1.88, τ = 0.5307 | k = 13.44, n= 1.87, τ = 0.5563 |
| Kashchiev | k = 269.7, n= 1, τ = 0.3635 | k = 19.31, n= 1.93, τ = 0.7589 | |
| Hill | k = 0.5193, n= 5.96 | k = 0.5421, n= 4.92 | |
| Logistic | k = 27.13, t0 = 0.5453 | k = 7.514, t0 = 0.5692 | |
| GSM | k = 27.71, t0 = 0.5453, m = 1.08 | k = 9.197, t0 = 0.5692, m = 0.368 |
| Protein | Model | t1/2 (days) | tmax (days) | W1/2 (days) | (dX/dt)max (μm/day) | Model | t1/2 (days) | tmax (days) | W1/2 (days) | (dX/dt)max (μm/day) |
|---|---|---|---|---|---|---|---|---|---|---|
| Hanging Drop (HD) | Langmuir-Blodgett (LB) | |||||||||
| Lysozyme | Hill | 1.68 | 1.59 | 0.949 | 404 | Hill | 1.14 | 0.859 | 1.22 | 617 |
| Thaumatin | Logistic | 4.18 | 4.18 | 4.15 | 18.3 | GSM | 3.75 | 3.83 | 4.47 | 43.7 |
| Ribonuclease A | Hill | 0.937 | 0.315 | 1.09 | 40.8 | Hill | 0.809 | 0.414 | 0.992 | 52.2 |
| Proteinase K | Logistic | 0.545 | 0.545 | 0.13 | 381 | Hill | 0.542 | 0.498 | 0.366 | 177 |
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Pechkova, E.; Speranza, F.M.; Ghisellini, P.; Rando, C.; Barbaro, K.; Eggenhöffner, R. Hybrid Kinetic Modelling of Protein Crystallization: Hanging Drop and Langmuir–Blodgett Conditions. Crystals 2025, 15, 857. https://doi.org/10.3390/cryst15100857
Pechkova E, Speranza FM, Ghisellini P, Rando C, Barbaro K, Eggenhöffner R. Hybrid Kinetic Modelling of Protein Crystallization: Hanging Drop and Langmuir–Blodgett Conditions. Crystals. 2025; 15(10):857. https://doi.org/10.3390/cryst15100857
Chicago/Turabian StylePechkova, Eugenia, Fabio Massimo Speranza, Paola Ghisellini, Cristina Rando, Katia Barbaro, and Roberto Eggenhöffner. 2025. "Hybrid Kinetic Modelling of Protein Crystallization: Hanging Drop and Langmuir–Blodgett Conditions" Crystals 15, no. 10: 857. https://doi.org/10.3390/cryst15100857
APA StylePechkova, E., Speranza, F. M., Ghisellini, P., Rando, C., Barbaro, K., & Eggenhöffner, R. (2025). Hybrid Kinetic Modelling of Protein Crystallization: Hanging Drop and Langmuir–Blodgett Conditions. Crystals, 15(10), 857. https://doi.org/10.3390/cryst15100857

