Research

Mathematical modeling and simulation predict the threshold of polyubiquitin size for optimal NF-kB activation

Mohammad Mobashir 1,*

1 Department of Biomedical Laboratory Science, Faculty of Natural Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway.

* Correspondence: mmobashir@jacobs-alumni.de (M.M.)

Citation: Mobashir, M. Mathematical modeling and simulation predict the threshold of polyubiquitin size for optimal NF-kB activation. Glob. Jour. Bas. Sci. 2025, 1(10). 1-6.

Received: April 29, 2025

Revised: August 11, 2025

Accepted: September 01, 2025

Published: September 02, 2025

doi: 10.63454/jbs20000056

ISSN: 3049-3315

Volume 1; Issue 11

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Abstract: The nuclear factor-κB (NF-κB) signaling cascade plays an essential role in mediating innate and adaptive immune responses, inflammatory processes, and cell survival. Signal transduction to the IκB kinase (IKK) complex is critically regulated by the attachment of polyubiquitin chains, specifically Lys63-linked modifications, to key intermediates such as receptor interacting protein 1 (RIP1) and TNF receptor–associated factors (TRAFs). However, despite extensive biochemical characterization, the precise quantitative relationship between ubiquitin chain length and the efficiency of NF-κB activation is not well established. To address this, I developed a detailed mechanistic model based on ordinary differential equations that explicitly integrates the dynamics of ubiquitin chain assembly and disassembly. Simulations and sensitivity analysis of this model indicate that a minimum threshold chain length is required for effective IKK activation and subsequent NF-κB nuclear translocation. Specifically, our predictions suggest that chains comprising fewer than approximately two to six ubiquitin subunits are insufficient to propagate a robust signal, while chains extending beyond this optimal range yield progressively diminished activation due to compensatory feedback mechanisms. This work offers a quantitative framework for understanding how the ubiquitin code is interpreted within the NF-κB pathway and proposes novel experimental approaches for fine-tuning innate immune signaling. This study summarizes the overall findings from the in-silico simulations of NF-κB interactions with different chains of PUB. The results clearly demonstrate consistent and optimal activation patterns of NF-κB across all examined signaling routes, indicating that NF-κB activation occurs independently of the specific upstream pathway, including TNFR, TLR, and TCR pathways.

Keywords: NF-κB signaling; polyubiquitination; mathematical modeling; systems biology; signal transduction; ubiquitin chain length; computational simulation; IKK activation

1. Introduction

The NF-κB family of transcription factors governs essential gene expression programs central to immune responses, inflammatory reactions, and cellular survival [1]. In an unstimulated state, NF-κB dimers are retained within the cytoplasm through their interaction with inhibitory IκB proteins. Activation of cell surface receptors, including those for TNF-α, IL-1β, and Toll-like receptor ligands, initiates a cascade that culminates in the activation of the IκB kinase (IKK) complex. This leads to the phosphorylation and proteasomal degradation of IκB, liberating NF-κB to translocate to the nucleus and drive target gene transcription [2–4].

Ubiquitination serves as a pivotal regulatory layer within this signaling pathway. This post-translational modification involves the covalent attachment of ubiquitin moieties to substrate proteins [5]. The assembly of Lys63-linked polyubiquitin chains on central adaptor proteins, such as RIP1, is particularly critical for the recruitment and activation of downstream kinase complexes, including TAK1 and IKK [6–9]. Emerging evidence underscores that the length and specific linkage architecture of these ubiquitin chains are key determinants of signaling specificity and strength [10–14]. Despite this recognition, precise quantitative parameters—such as the minimal chain length required for efficient signal propagation—are not well defined.

Mathematical modeling offers a powerful approach to interrogate the quantitative logic of complex biological systems and to predict non-intuitive dynamic behaviors [15–19]. Existing computational models have successfully captured oscillatory NF-κB dynamics [20–23], delineated feedback mechanisms [24–27], and explored pathway cross-talk [28–30]. However, few models have explicitly integrated the detailed kinetics of ubiquitin chain assembly and disassembly [31–33]. 

To address this gap, I constructed a comprehensive ordinary differential equation (ODE) model of a canonical NF-κB activation pathway. This model explicitly incorporates the dynamics of ubiquitin chain elongation on key signaling intermediates, their recognition by ubiquitin-binding domains, and the consequent activation of the IKK complex. Through systematic simulation and sensitivity analysis, I identify a critical threshold in polyubiquitin chain length necessary for effective signal transduction. Our findings provide a quantitative framework for deciphering how the ubiquitin code is interpreted in immune signaling and may inform strategies for the therapeutic modulation of the NF-κB pathway.

2. Methods

2.1. Model preparation: A deterministic ordinary differential equation (ODE) model was developed to simulate the core molecular events in canonical NF-κB signaling, as illustrated in Figure 1. The model framework encompasses the sequential steps from receptor activation—initiated here by TNF-α binding to TNFR1—through the  downstream nuclear translocation of NF-κB. Key processes explicitly represented include the recruitment of adaptor proteins (TRADD, TRAF2/5), the K63-linked polyubiquitination of RIP1, the subsequent assembly and activation of the TAK1 and IKK complexes, the phosphorylation and proteasomal degradation of IκBα, and the nuclear import of liberated NF-κB dimers [34–38]. To quantitatively investigate the role of ubiquitin chain length, the process of polyubiquitin chain assembly was modeled as a stepwise, enzyme-catalyzed reaction sequence. Substrates such as RIP1 were represented by distinct chemical species for each possible chain length, from the unmodified state (U0) up to a maximum length (Un) [39–42]. The model specifically assumes the formation of K63-linked chains, consistent with their established role in this pathway [43–45]. Reaction rates were formulated using mass-action kinetics. Kinetic parameters and initial concentrations were sourced from published biochemical and cell-biological studies where available [46–49]. Remaining parameters were systematically estimated and refined through computational fitting to published experimental time-course  data for IKK activity and NF-κB nuclear localization [50,51].

Figure 1. A model to show the role of polyubiquitin chain length in modulating the NF-kB activation.

2.2. Simulation and analysis: All simulations were executed in MATLAB R2023b (The MathWorks, Inc.) using the stiff ODE solver ODE15s to ensure numerical stability. To evaluate the influence of model parameters on signaling output, a global sensitivity analysis was performed using the calculation of partial rank correlation coefficients (PRCCs). This method identifies parameters that exert critical control over peak nuclear NF-κB concentration and response dynamics [52–54]. To probe for threshold behavior in ubiquitin chain length, simulations were conducted while systematically varying parameters controlling chain elongation rates or the availability of ubiquitin-conjugating machinery. The resulting impact on the time-integrated or peak level of nuclear NF-κB was quantified to define signaling efficacy.

2.3. Model validation: The predictive capacity of the model was assessed by comparing its outputs against independent experimental datasets not used during parameterization. These validation experiments involved genetic or pharmacological perturbations known to alter polyubiquitin chain dynamics, such as the overexpression or knockdown of specific ubiquitin ligases or  deubiquitinases (DUBs) [55–57].

3. Results

3.1. Model recapitulates core NF-κB signaling dynamics: Here, I prepared the mathematical model in which I used ODEs and evolutionary algorithm to evolve the kinetics of the target molecules for different sets of kinetic parameters (rate of reaction). Simulation of the model in response  to a TNF-α stimulus successfully reproduced the established sequence of canonical NF-κB activation. The output demonstrated the rapid assembly of K63-linked polyubiquitin chains on the adaptor protein RIP1, which  facilitated the recruitment and activation of the TAK1 and IKK kinase complexes. This led to the characteristic phosphorylation and subsequent degradation of the inhibitor IκBα, followed by the transient nuclear translocation of NF-κB. In Figure 2, I presented the plots for the concentrations of different chains in terms of number of ubiquitin at different time points. Under conditions of continuous receptor stimulation, the model generated transient patterns of nuclear PUB concentration, a dynamic behavior well-documented in prior experimental and computational studies.

Figure 2. Ubiquitin chains of different lengths (2-20) and the concentration of ubiquitin chains with respect to time.

3.2. A quantitative threshold in ubiquitin chain length governs signaling efficacy: After evaluating the patterns of PUB concentration at different time points, I performed the analysis of NF-kB activation patterns for different receptors from where it is activated. Thus, I could say that to interrogate the functional impact of ubiquitin chain size, I systematically varied the model parameters controlling the rate of chain elongation on RIP1. This analysis revealed a sharp and direct relationship between chain length and signaling output or threshold of NF-kB activation. Efficient activation of the IKK complex required polyubiquitin chains to reach a critical threshold of approximately four to six ubiquitin subunits (Figure 3). Simulations with chain lengths below this threshold resulted in insufficient multivalent scaffolding, leading to poor recruitment of the TAK1  complex and a markedly attenuated NF-κB response (Figure 3). Conversely, extending chains beyond this optimal range provided only minimal increases in signal amplitude, consistent with a model where the binding sites on downstream adaptor proteins become saturated.

Figure 3. Ubiquitin chains of different lengths (2-20) and the concentration of ubiquitin chains with respect to time.

Finally, it lead to the conclusions that the PUB chain length is crucial for that exert the strongest influence on the peak nuclear NF-κB concentration. This analysis highlighted the kinetic parameters associated with the ubiquitination machinery as primary determinants of signaling strength. Notably, parameters governing the initial priming and early elongation steps of ubiquitin chains had a greater impact on the final output than those controlling the later stages of chain extension.

4. Discussion

Our computational analysis reveals that efficient NF-κB activation is governed by a quantifiable threshold in polyubiquitin chain length. This prediction aligns with the established biochemical role of K63-linked chains as physical scaffolds that nucleate the assembly of high-order signaling complexes, such as those containing TAK1 and IKK. Chains shorter than the predicted threshold lack the multivalent binding sites necessary to stabilize these critical interactions, leading to attenuated signal propagation [7,10]. Conversely, extending chains beyond this optimal length provides diminishing returns, as the available binding sites on adaptor proteins become saturated, limiting any additional signaling benefit. 

This identified threshold mechanism offers novel insights into immune regulation and pathogenesis. Cellular responsiveness to stimuli could be dynamically tuned by enzymes that control chain length—namely, ubiquitin ligases that catalyze elongation and deubiquitinases (DUBs) that mediate trimming. Precise manipulation of these enzymes could thus shift cells into either hypo-responsive or hyper-responsive states. This framework provides a rationale for therapeutic strategies aimed at the ubiquitin system, suggesting that targeted modulation of specific ligases (e.g., cIAP1/2) or DUBs (e.g., CYLD, A20) could allow for the fine-tuning of NF-κB activity in contexts of chronic inflammation or cancer [21-22,58–63].

While our model successfully captures the core dynamics linking ubiquitin chain length to signaling output, it incorporates several  necessary simplifications. The current framework focuses exclusively on K63-linked polyubiquitination. However, other ubiquitin linkage types, such as Met1-linked linear chains, are known to play significant and distinct regulatory roles in NF-κB pathways [64–66]. Furthermore, the model employs a deterministic formulation and does not account for potential stochastic effects arising from low molecular copy numbers in single cells. 

In the end, I also summarised the overall findings of the in-silico simulation of NF-kB and different chains of the PUB in Figure 4. Here, it is clearly showing the optimal activation of week activation patterns if NF-kB irrespective of the NF-kB activation pathways (TNFR, TLR, TCR). More details of the exact implementation of the algorithm could be seen in the previous works [67-68]. Future iterations of this work will aim to address these limitations by integrating the dynamics of multiple ubiquitin linkage types and by developing a stochastic modeling framework to explore cell-to-cell variability.  These advancements will provide a more comprehensive and physiologically detailed understanding of how the ubiquitin code is interpreted to control inflammatory signaling.

Figure 4. Ubiquitin chains of different lengths (2-20) and the concentration of ubiquitin chains with respect to time.

5. Conclusions
In this study, I have established a quantitative link between the structural property of polyubiquitin chains and the functional output of a major immune signaling pathway. Through the development and analysis of a mechanistic mathematical model, I demonstrate that the NF-κB pathway exhibits a threshold dependence on K63-linked polyubiquitin chain length for effective signal transduction. Our simulations predict that chains must reach a critical minimum length of approximately two to six ubiquitin units to nucleate a stable signaling complex and trigger robust downstream activation. These findings provide a critical advance in deciphering the “ubiquitin code,” illustrating how a quantitative, non-binary attribute—chain length—is interpreted by the cell to gate the initiation of a transcriptional response. By mapping this relationship, our work moves beyond qualitative descriptions of ubiquitin’s role to a predictive, quantitative framework. 

This computational model serves as a foundational tool for exploring therapeutic strategies. It allows for the in silico screening of interventions targeting the ubiquitin machinery—such as specific E3 ligase inhibitors or deubiquitinase modulators—and predicts their potential to rewire signaling thresholds in pathological states like chronic inflammation or cancer. Future integration of additional regulatory layers and linkage types will further refine this framework, enhancing its utility for both fundamental discovery and translational innovation in immune signaling.

In conclusion, this study provides a comprehensive synthesis of the in-silico simulation results examining NF-κB activation in the context of its interactions with different chains of PUB. Across all computational models and simulation conditions, NF-κB exhibited robust and reproducible activation dynamics, highlighting a conserved regulatory behavior. Notably, the activation patterns remained consistent regardless of the upstream signaling cascade involved, including tumor necrosis factor receptor (TNFR), Toll-like receptor (TLR), and T-cell receptor (TCR) pathways. These findings suggest that NF-κB functions as a central signaling hub capable of integrating diverse extracellular cues into a unified transcriptional response. The observed pathway-independent activation underscores the intrinsic stability and resilience of the NF-κB signaling network, likely supported by extensive feedback regulation and cross-talk among signaling components. Furthermore, the interaction of NF-κB with different PUB chains did not significantly alter its activation efficiency, indicating that PUB-mediated modulation may fine-tune downstream responses rather than dictate pathway specificity. Overall, this computational analysis enhances our understanding of NF-κB signaling robustness and supports the concept that NF-κB activation is governed more by network architecture than by individual receptor-specific inputs. These insights provide a valuable framework for future experimental validation and may inform therapeutic strategies aimed at modulating NF-κB activity in inflammatory, immune, and cancer-related contexts.

Author Contributions: Conceptualisation, M.M.; software, M.M.; investigation, M.M.;  writing—original draft preparation, M.M.; writing—review and editing, M.M.; visualisation, M.M.; supervision, M.M.; project administration, M.M. The author has read and agreed to the published version of the manuscript.

Funding: Not applicable.

Acknowledgments: I am grateful to the Department of Biomedical Laboratory Science, Faculty of Natural Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway for providing us all the facilities to carry out the entire work.

Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: All the related data are supplied in this work or have been referenced properly.

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