Dry Lab

When we think about the Dux4-DBD vs Dux4-FL competitive inhibition and ways to apply it as a therapeutic for different patients, multiple challenges arise. First, Dux4-FL concentrations are highly variable between patients and even within individual cells.[1] Additionally, muscle fibers are syncytial, meaning they contain multiple nuclei, with Dux4-FL potentially bursting in a low fraction of these nuclei. This variability complicates the definition of the necessary amount of Dux4-DBD needed to counteract aberrant expression. Furthermore, the treatment gene will be delivered to an estimated fraction of nuclei independently, adding another layer of complexity. Mapping out a full range of possible inhibition scenarios is essential, but doing so solely through experimental approaches would be challenging due to time and cost constraints. Our team devised this as a challenge: experimental data alone may not provide a comprehensive understanding of the competitive dynamics between [FL] and [DBD].

So, our central questions are: What is the optimal amount of Dux4-DBD needed to set a quantitative target inhibition level to inhibit DUX4 target gene transcription effectively, thereby preventing cell death and slowing the progression of FSHD? How can we determine the precise amount of [DBD] required to outcompete [FL] in each patient to achieve this inhibition?

That’s why we decided to develop our models to attempt to aid in understanding and answering these questions. Modeling is crucial to answer these questions because it allows us to simulate a wide range of conditions, predict outcomes across different scenarios, and refine our understanding of the system. By analyzing the patterns of interactions both over space and time, we can gain a more robust understanding of the safety and applicability of dux. as a therapeutic approach.

Transcription Factor Binding Model

We started from the very crux of the project: competitive inhibition. When examining a single cell, how can we accurately gauge the quantitative target inhibition level? Modeling at the single cellular level is crucial for several reasons. First, it allows us to understand the system at a fundamental level, providing insights into how DUX4-DBD competes with DUX4-FL on a cellular scale. This is essential for verifying that our experiments are working as intended and for refining our models to accurately predict outcomes. By studying the system in a single cell, we can ensure that our approach is sound before scaling it up, which is critical for developing effective and safe treatments.

To understand this, we modeled the transcription factor dynamics that influence the behavior of DUX4-DBD at the binding sites that DUX4-DBD and DUX4-FL bind to. The model uses Dux4-FL concentration as a constant input, reflecting its variability among patients, and varies the Dux4-DBD concentrations (up to 1000x higher) to determine the optimal concentration required for achieving 99.9% inhibition of Dux4-FL. We then conducted wet lab experiments to measure transcriptional output using reporters in HEK cells, comparing these results to the model’s predictions. By validating the model against experimental data, we aim to interpret the wet lab data based on our predicted mechanism of action to find the optimal DBD-to-FL ratio needed for effective inhibition of FL-driven transcription.

To understand this, we first modeled the transcription factor dynamics at a single binding site, using the biophysical competition equation:

Zhang, Y., Ho, T. D., Buchler, N. E., & Gordân, R. (2021). Competition for DNA binding between paralogous transcription factors determines their genomic occupancy and regulatory functions. Genome Research, 31(7), 1216–1229. https://doi.org/10.1101/gr.275145.120

In our case: [TF1] represents DUX4-FL concentration. [TF2] represents DUX4-DBD concentration. KD is the dissociation constant, assumed to be the same for both DUX4-DBD and DUX4-FL, with a value of 1.4 μM.[3]

This equation models the competitive binding dynamics between DUX4-DBD and DUX4-FL for the same binding site. By applying this model, we varied the DUX4-DBD concentrations (up to 1000x higher) against constant DUX4-FL concentrations to determine the optimal concentration for complete DUX4-FL inhibition.

Next, we extended this model to multiple binding sites, as it's important to consider that DUX4 can bind to several sites on DNA. A study from the Kyba Lab at the University of Minnesota, titled “DNA-binding sequence specificity of DUX4,” found that when DUX4-FL binds to several of these sites, it works together in a way that boosts its overall activity.[8] This means that the more binding sites DUX4-FL connects to, the stronger its effect on gene expression, which is something we considered in our model.

Evaluating reporters with multiple DUX4 binding sequences. a Dose-dependent activity of reporters with 1–24 copies of the CT motif at 100 ng of DNA/well. Note the logarithmic scale. b Dose-dependent activity of the same reporters at 4 ng/well. Synergy is demonstrated by the presence of activation with the 2× reporter but its absence with the 1× reporter at this concentration, as well as the continued increase in activity as copy number increases to 24×. c Transcriptional activity of reporters with two CT motifs spaced at different distances (indicated), compared to a reporter with a single CT motif. Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4730607/

Evaluating reporters with multiple DUX4 binding sequences. a Dose-dependent activity of reporters with 1–24 copies of the CT motif at 100 ng of DNA/well. Note the logarithmic scale. b Dose-dependent activity of the same reporters at 4 ng/well. Synergy is demonstrated by the presence of activation with the 2× reporter but its absence with the 1× reporter at this concentration, as well as the continued increase in activity as copy number increases to 24×. c Transcriptional activity of reporters with two CT motifs spaced at different distances (indicated), compared to a reporter with a single CT motif. Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4730607/

In the overall model we assumed the following: The dissociation constant of Dux4-DBD and Dux4-FL binding to a DNA binding siteare the same. First two assumptions are safe because the DNA binding domain of FL and DBD are identical. Dux4-DBD and Dux4-FL are competing for the same binding site. For a multiple binding site promoter, binding at any of the DUX4 consensus binding site has an equivalent effect on the promoter output (also assume that binding at a site is independent of binding at adjacent sites). Both proteins have the same half-life.

Output of Transcription Binding Model

The [FL] concentrations delimited were:
FL_concentration_high = 1e-3 M, where FL concentration >> kD
FL_concentration_mid = 1e-6 M, where FL concentration ~= kD
FL_concentration_low = 1e-9 M, where FL concentration << kD

The [DBD] concentration that achieves 99.9% inhibition of Dux4-FL based on the lab data is 187.38 µmol/L. This high concentration is largely due to the elevated [FL] levels in the transient transfection system, which are much higher than those typically observed in physiological conditions. In clinical contexts, transcription factors generally operate at much lower concentrations, often in the nanomolar to low micromolar range. Therefore, for therapeutic design, we should target a [DBD] concentration that aligns with the [FL] levels typically observed in patients, which are significantly lower. This adjustment is essential to developing a therapeutic approach that is both effective and safe within a physiological context.

After this initial prediction, we plotted the wet lab data on top of the model to validate the predicted system.

The data we used was normalized by trial, we had 3 trials and averaged all 3 to make a 4th dataset.

While the model generally aligns with the trend observed in the experimental results, particularly when using the normalized dataset, there is a notable fit at a DBD-FL concentration of 0.000013 mol/L, which corresponds well with the experimental data. However, discrepancies become more apparent when using non-normalized data, especially around specific DBD concentrations, such as 1e-6 mol/L. These discrepancies suggest that while the model captures the overall binding dynamics, there may be additional factors or nuances in the biological system not fully accounted for by the model. Adjusting certain assumptions, such as the exact binding affinity or the influence of other molecular interactions, could potentially explain the observed offset between the model and experimental data in these cases.

At higher DBD concentrations, our model predicted significant inhibition of transcriptional output, which we confirmed experimentally. However, we noticed some deviations at lower concentrations. To address this, we measured the protein half-life of both DBD and FL in FSHD-affected cells and assessed promoter strength under different conditions. These experiments revealed that differences in protein stability and promoter strength were influencing our model's accuracy. Using this data, we refined our model parameters, leading to more accurate predictions of the optimal DBD concentration needed to inhibit DUX4 target gene transcription effectively.

After understanding the range of necessary [DBD] to outcompete [FL] in a single cell, another important question arises: How can we optimize the reach of [DBD] in different myonuclei? Can we assure that the therapeutic will reach the nucleus expressing Dux4-FL, considering that its expression is low amongst FSHD patients?

Myotube Diffusion Model

The primary output of the binding model, as observed, is the optimal concentration of Dux4-DBD that achieves complete inhibition of Dux4-FL-driven transcription. However, this optimal Dux4-DBD concentration is not just an isolated finding–it is a crucial input for our subsequent model, where we extend the analysis from the single-cell level to a higher level system dynamics over time.

So, after modeling what can be compared with wet lab HEK cells experiments, and understanding more the spatial dynamics, we wanted to zoom out into the dynamic interaction of FL and DBD over time. How does the dynamic interaction between DUX4-FL and varying concentrations of DBD over time influence the inhibition of FL-driven transcription, and what is the optimal ratio needed to effectively suppress [FL] expression in different cellular contexts?

This model utilizes Biopython and Matplotlib to simulate the diffusion of the DNA-binding domain (DBD) protein within muscle fibers, offering insights into its behavior alongside known fluorescent proteins. The process begins by analyzing the DBD protein sequence, sourced from a public protein database, to compute its molecular weight using Biopython's ProteinAnalysis. This molecular weight is compared with those of well-known fluorescent proteins such as mCherry, tdTomato, and DsRed, providing a reference for further analysis.

The diffusion coefficient of the DBD protein is then estimated using linear interpolation, based on the diffusion rates of mCherry and tdTomato. This approximation helps predict the DBD protein's rate of movement within the muscle fibers relative to the fluorescent proteins, given their molecular sizes. Following this, the model performs a numerical simulation to solve the diffusion equation along a muscle fiber. The simulation computes the concentration profile of the DBD protein over time, providing a snapshot of how the protein spreads across the fiber.

Additionally, the model calculates steady-state concentration profiles for DBD and the fluorescent proteins using an exponential decay function, showing the relative concentration of these proteins along the length of the fiber. The results are visualized with Matplotlib, plotting diffusion profiles for DBD, mCherry, tdTomato, and DsRed, illustrating how DBD compares in terms of diffusion dynamics. This comprehensive model combines computational biology and numerical simulation, offering valuable insights into the diffusion behavior of the DBD protein in a biological context.

Ordinary Differential Equation (ODE) Model

Building on the insights from the Transcription Factor Binding Model, The ODE (Ordinary Differential Equation) model simulates the dynamic interaction between Dux4-FL and Dux4-DBD over time, the ODE model simulates how the dynamic interaction between Dux4-FL and Dux4-DBD evolves over time across a population of cells. Here, the optimal Dux4-DBD concentration identified in Model 1 is used as a key input to assess its effectiveness in suppressing Dux4-FL-driven transcription over a longer period and across different cellular contexts. This model answers critical questions such as:

• How does the timing of [DBD] administration affect its effectiveness?
• At what concentration does DBD most effectively suppress FL-driven transcription for a longer period of time?

Overall, it takes another step back to a bigger picture, providing another perspective on the optimal amount of [DBD] required to effectively suppress FL-driven transcription and assess its efficiency as an inhibitor over time. This is particularly useful to assess efficiency of a future therapeutic, as we can analyze the behavior of this competition after a specific time frame.

The model, which applies stochastic gene expression, accounts for random promoter activation, transcriptional bursting, and mRNA and protein degradation. The model then runs over a defined time frame (48 hours) and applies an ODE solver to track the changes in cell states (S: healthy, E: exposed, I: infected, R: recovered, D: dead).[4] This is a compartment model, which describes the states and transitions of FSHD single myocytes based on their DUX4 and DUX4 target gene expression.

Cowley et al. “An in silico FSHD muscle fiber for modeling DUX4 dynamics and predicting the impact of therapy” https://elifesciences.org/articles/88345

We then applied the Poisson-Beta distribution to estimate the expression levels of DUX4 isoforms based on simulated mRNA transcripts.[5] The Poisson distribution handles count data (in this case, the number of mRNA transcripts), while the Beta distribution models proportions (the relative expression levels of Dux4 isoforms). Additionally, we used the dynamic competition between Dux4-FL and Dux4-DBD as another factor to inform a dynamic infection rate throughout the cells.

The parameters incorporated were: Base expression rates of Dux4-FL and Dux4-DBD (0.39 TPM and 0.25 TPM respectively), [6] Optimal Dux4-DBD concentration derived from the transcription factor binding model, Posterior parameters (alpha and beta) derived from a Poisson-Beta distribution model that estimates the expression levels of DUX4 isoforms based on simulated mRNA transcripts, Compared to the paper, the model dynamically adjusts the infection rate, which is assumed to serve as a proxy for disease progression, influenced by the ratio of Dux4-DBD to Dux4-FL.

We also sourced the ODEs from this same paper, which describe the changes of these five states over time:

Cowley, M. V., Pruller, J., Ganassi, M., Zammit, P. S., & Banerji, C. R. (2023). An in silico FSHD muscle fiber for modeling DUX4 dynamics and predicting the impact of therapy. eLife, 12. https://doi.org/10.7554/elife.88345

As we moved forward, we realized that the “one-size-fits-all” approach of ODEs might not be the optimal representation of FSHD. As each individual has different progressions, it is crucial to create predictions based on a personalized starting point, and also account for the inherent uncertainty/randomness of gene expression.

Markov Model

The Markov model provides a framework for understanding the dynamic interaction between DUX4-FL and varying concentrations of DBD by modeling the system as a series of discrete states that a cell can transition through over time with more insights compared to the ODE. By estimating the probabilities of transitions between these states—such as from Susceptible (no DUX4-FL) to Exposed (DUX4-FL mRNA present), and from Infected (DUX4-FL mRNA and protein present) to Resigned (lingering DUX4-FL protein) or Dead (cell death due to DUX4-FL toxicity)—the Markov model captures the stochastic nature of gene expression and the discrete events that lead to either the suppression or the persistence of FL-driven transcription.

Cowley, M. V., Zammit, P. S., & Banerji, C. R. S. (n.d.). Quantifying anti-DUX4 therapy for facioscapulohumeral muscular dystrophy. bioRxiv (Cold Spring Harbor Laboratory). https://doi.org/10.1101/2024.08.14.607973

Through the simulation of these transitions, the Markov model can predict the optimal ratio of DBD to FL that minimizes the likelihood of DUX4-FL driving harmful transcriptional activity, even in the face of cellular variability. This approach provides critical insights into how different DBD concentrations affect the overall outcome in various cellular contexts, helping to tailor therapeutic strategies to effectively inhibit FL expression across different patients.

What is the likelihood of successful FL inhibition under varying DBD concentrations? The Markov model can predict the probabilities of different outcomes based on DBD concentration, offering insights into the most effective strategies for different cellular contexts.

How does cellular variability impact the effectiveness of DBD treatment? By simulating the range of possible cellular states, the Markov model helps to tailor therapeutic strategies to effectively inhibit FL expression across different patients.

Future Directions

We plan to further investigate the potential of using MRI as a less invasive method compared to traditional biopsies for obtaining critical information on DUX4-FL and DBD interactions. Recognizing the emergence of MRI as a promising technique, we developed an MRI image preprocessing pipeline that can enhance the accuracy and utility of these analyses, providing a more patient-friendly approach to monitoring disease progression and treatment efficacy.

Additionally, we will explore the impact of DBD on the overall transcriptional landscape, particularly focusing on downstream genes influenced by DBD activity. This will help determine whether the current levels of DBD administration are optimal or if adjustments are needed. Understanding these effects is crucial to ensure that the therapeutic use of DBD is both effective in inhibiting FL-driven transcription and safe, without causing unintended alterations in gene expression.