Mathematical Structure of RelB Dynamics in the NF-κB Non-Canonical Pathway
Abstract
:1. Introduction
2. Materials and Methods
2.1. Negative Feedback from IB Is Essential for the Self-Sustained Oscillation of RelB
- (a) Certain elements activated via nuclear p50/RelB switched the activation state of IB (i.e., whether IB can bind to RelB).
2.2. Full Non-Canonical Pathway Model
3. Results
3.1. Parameter Calibrations
3.2. Three Types of RelB Dynamics Are Dependent on the Concentrations of Total and
4. Discussion
4.1. Dependency of RelB Dynamics on Concentrations of and
4.2. Quantitative Consideration of the Formation of
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Detailed Analysis of Time Evolutions of with the 14 Parameters
Appendix B. Bifurcation Analysis
Appendix B.1. Algorithm for Finding the Steady State
Appendix B.2. Linearization and Eigenvalue Problems
- If all the eigenvalues of the Jacobian have negative real parts, then the equilibrium point () is asymptotically stable.
- If any eigenvalue of the Jacobian has a positive real part, then becomes unstable.
- When complex eigenvalues appear in the Jacobian, X approaches while oscillating.
totalRelB [M] | totalIkBa [M] | [rad/min] | Equivalent Period: 2 [h] |
---|---|---|---|
0.8 | 2.5 | 0.0590 | 1.77 |
0.4 | 2.5 | 0.0316 | 3.32 |
0.4 | 5.0 | 0.0286 | 3.66 |
0.4 | 12.5 | 0.0636 | 1.65 |
Appendix C. Experimental Methods and Data Details
- Cell preparation: MEF (mouse embryonic fibroblast) cells stably expressing RelB-Venus were used. These cells were modified with a retroviral vector in which the nuclear localization signal (NLS) from the nuclear fluorescent protein EYFP-Nuc was fused to the 3’ end of mCherry [20].
- Antibody treatment: Cells were treated with 0.3 g/mL anti-LTR antibody to induce RelB nuclear translocation. This treatment activates the non-canonical NF-B pathway [20].
- Time-lapse imaging: Cells were observed using time-lapse imaging, and the fluorescence intensity in the nucleus and the entire cell was measured. These data show the nuclear fluorescence intensity of RelB, particularly the time evolution following stimulation with 0.3 g/mL anti-LTR antibody. The experiment observed patterns of RelB nuclear translocation and oscillation after stimulation [20].
Appendix C.1. Empirical Evidence for the Non-Canonical NF-κB Pathway
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Value | Molecule p100KO-M | NC-FM |
---|---|---|
p52/RelB | p52/RelB | |
IB | IB | |
IB/p52/RelB | IB/p52/RelB | |
p52/RelBi | p52/RelBi | |
IB | IB | |
IB/p52/RelBi | IB/p52/RelBi | |
unknown | unknown | |
inactive IB | p100/RelB | |
- | p100 | |
- | p100/RelBi | |
- | p100i | |
- | p100 mRNA | |
- | active IKK | |
- | inactive IB |
Pathway | Description | |
---|---|---|
(i) | p50/RelB → p50/RelBi | Cytoplasmic p50/RelB moves into the nucleus |
(ii) | IB IB | IB shuttles between the nucleus and cytoplasm |
(iii) | p50/RelB/IB → p50/RelB/IB | Nuclear p50/RelB exports to the cytoplasm by binding to IB |
(iv) | p50/RelB/IB → p50/RelB | IB of p50/RelB/IB degrades in the cytoplasm |
Symbol | Value | Unit | Description |
---|---|---|---|
0.2 * | 1/min | p52/RelB nuclear import | |
30 * | 1/M min | p52/RelB association | |
0.00678 ‡ | 1/min | IB degradation | |
0.018 ‡ | 1/min | IB nuclear import | |
0.2448 ‡ | 1/M min | IB activation | |
0.012 ‡ | 1/min | IB nuclear export | |
0.1 * | M/min | p52/RelB-dependent (unknown factor) transcription | |
0.0168 ‡ | 1/min | IB mRNA degradation | |
0.012 * | 1/min | IB/p52/RelB nuclear export | |
0.1 * | 1/min | IB of IB/NF-B dissociation | |
50 * | 1/M min | p52/RelB replacement with p100/RelB | |
0.001 * | 1/min | p100/RelB nuclear import | |
0.01 * | 1/min | p100/RelB nuclear export | |
0.01 * | 1/min | p100 nuclear import | |
0.01 * | 1/min | p100 nuclear export | |
0.1 * | 1/min | p100 protein translation | |
0.1 * | 1/min | p100 degradation | |
0.1 * | 1/min | p100 mRNA degradation | |
0.001 * | M/min | p100 mRNA transcription | |
0.5 * | 1/M min | p100 processing | |
0.0072 ‡ | 1/min | IKK degradation | |
0.4 * | M | Total amount of RelB in a cell in the oscillation event | |
2.5 * | M | Total amount of IB in a cell in the oscillation event | |
0.1 * | M | Total amount of activeIKK in a cell |
Symbol | PAV | EMV | Unit | Description |
---|---|---|---|---|
0.2 * | 5.4 † | 1/min | p52/RelB nuclear import | |
30 * | 51 † | 1/M min | p52/RelB association | |
0.00678 ‡ | as left | 1/min | IB degradation | |
0.018 ‡ | as left | 1/min | IB nuclear import | |
0.2448 ‡ | as left | 1/M min | IB activation | |
0.012 ‡ | as left | 1/min | IB nuclear export | |
0.1 * | 0.01 * | M/min | p52/RelB-dependent (unknown factor) transcription | |
0.0168 ‡ | as left | 1/min | IB mRNA degradation | |
0.012 * | as left | 1/min | IB/p52/RelB nuclear export | |
0.1 * | 0.032 † | 1/min | IB of IB/NF-B dissociation | |
50 * | 0.13 † | 1/M min | p52/RelB replacement with p100/RelB | |
0.001 * | 0.028 † | 1/min | p100/RelB nuclear import | |
0.01 * | 0.42 † | 1/min | p100/RelB nuclear export | |
0.01 * | 0.045 † | 1/min | p100 nuclear import | |
0.01 * | 0.012 † | 1/min | p100 nuclear export | |
0.1 * | 0.5 # | 1/min | p100 protein translation | |
0.1 * | 0.4 # | 1/min | p100 degradation | |
0.1 * | 1.6 × # | 1/min | p100 mRNA degradation | |
0.001 * | 1.9 × # | M/min | p100 mRNA transcription | |
0.5 * | 4.2 # | 1/M min | p100 processing | |
0.0072 ‡ | as left | 1/min | IKK degradation | |
0.4 * | 0.08 $ | M | Total amount of RelB in a cell in the oscillation event | |
2.5 * | 0.6 * | M | Total amount of IB in a cell in the oscillation event | |
0.1 * | as left | M | Total amount of activeIKK in a cell |
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Umegaki, T.; Hatanaka, N.; Suzuki, T. Mathematical Structure of RelB Dynamics in the NF-κB Non-Canonical Pathway. Math. Comput. Appl. 2024, 29, 62. https://doi.org/10.3390/mca29040062
Umegaki T, Hatanaka N, Suzuki T. Mathematical Structure of RelB Dynamics in the NF-κB Non-Canonical Pathway. Mathematical and Computational Applications. 2024; 29(4):62. https://doi.org/10.3390/mca29040062
Chicago/Turabian StyleUmegaki, Toshihito, Naoya Hatanaka, and Takashi Suzuki. 2024. "Mathematical Structure of RelB Dynamics in the NF-κB Non-Canonical Pathway" Mathematical and Computational Applications 29, no. 4: 62. https://doi.org/10.3390/mca29040062