Single-Cell Protein-mRNA Modeling

Statistical modeling of the relationship between mRNA and protein abundances at single-cell resolution using XX distributions and log-normal noise models.

Overview

This project models protein abundance from mRNA counts in single cells by:

1. Fitting negative binomial (NB) distributions to mRNA count data
2. Simulating protein levels using a log-normal multiplicative noise model
3. Jointly optimizing mRNA and protein parameters to match observed distributions

The core model assumes:

P = (c * m + ε) * LogNormal(0, σ)

where m is mRNA count (NB distributed), c is a scaling factor, ε prevents log(0), and σ captures biological/technical noise.

Repository Structure

├── models/
│   ├── APPROACHES.md           # Modeling approaches and equations
│   └── Model_V1.ipynb          # Main analysis notebook (Google Colab)
├── preprocessing/
│   └── preproc_and_filter.R    # R preprocessing script for protein and mRNA data
├── dat/                        # Local data directory (optional)
└── requirements.txt            # Python dependencies

Data Access

Raw sequencing data is available on GEO: GSE244215

Processed data for modeling is stored on Google Drive: Data Folder

Google Drive Structure

This data is all for 1 cell type, basal cells (see preproc_and_filter.R)

Drive/
└── Protein_RNA_Modeling/
    ├── raw/           # mRNA counts, Raw protein data, metadata for both
    ├── processed/     # Cleaned/filtered protein data for our modeling purposes
    ├── results/       # Model outputs and figures
    └── notebooks/     # Working copies of Colab notebooks

Notebook Links

• GitHub source links (used by the Colab badge) will break if files are moved/renamed in the repo — update the badge URL if you reorganize.
• Google Drive links are based on file ID, not path — notebooks can be moved freely within Drive without breaking share links.

Data Loading (Built into Notebook)

The notebook includes a data loading cell that mounts Google Drive and loads the data. Just update the DATA_PATH variable to point to your data folder.

Quick Start

1. Open Model_V1.ipynb in Google Colab
2. Run the first cell to mount Google Drive
3. Update DATA_PATH to point to your data folder
4. Run cells sequentially

Core Functions

Function	Description
fit_nb_mle(x)	Fit negative binomial to mRNA counts via MLE
simulate_protein_log2fc_from_mrna_nb()	Simulate protein log2FC from mRNA model
fit_sigma_log_to_protein()	Grid search for optimal noise parameter σ
joint_fit_nb_and_sigma()	Joint optimization of NB and noise parameters

Dependencies

Python (Colab):

• numpy
• scipy
• matplotlib

R (preprocessing):

• Seurat
• stringr
• seqinr
• dplyr

Usage Example

# Fit NB to mRNA counts
mu_hat, r_hat, _ = fit_nb_mle(x_mrna)

# Find optimal sigma to match observed protein distribution
best_sigma = fit_sigma_log_to_protein(y_obs, mu_hat, r_hat)

# Or jointly optimize all parameters
result = joint_fit_nb_and_sigma(
    x_mrna,
    y_obs,
    t_half_m_hours=2.0,   # mRNA half-life
    t_half_p_hours=24.0   # protein half-life
)

License

MIT