Inherited or produced? Inferring protein production kinetics when protein counts are shaped by a cell's division history

This paper addresses the challenge of inferring protein production kinetics in dividing cells by overcoming the intractability of standard likelihood methods caused by non-Markovian division history effects, utilizing conditional normalizing flows to reveal that the yeast *glc3* gene exhibits mostly inactive, brief, and transient expression under nutrient stress.

The Gist

The Big Problem: The "Family Heirloom" Confusion

Imagine you are a detective trying to figure out how fast a factory is producing a specific toy (a protein). You walk into the factory and take a snapshot of the workers holding these toys.

In a normal factory, if you see a worker holding 10 toys, you might guess they made 10 toys recently. But in a cell, things are weird. When a cell divides (splits into two), it doesn't throw away its toys. Instead, it passes them down to its children, like a family heirloom.

So, if you see a cell holding 100 toys, it might be because:

1. The factory is running at full speed right now (high production).
2. OR, the factory has been quiet for a long time, but the cell just inherited a huge pile of toys from its great-grandmother, grandmother, and mother.

The Mistake: If you ignore the "family history" (the cell's division history), you will think the factory is super busy when it's actually sleeping. This is the problem the authors faced: How do you tell the difference between a toy made today and a toy inherited from yesterday?

The Old Way vs. The New Way

The Old Way (The "Perfect Math" Approach):
Scientists usually try to write a perfect math equation (a likelihood) to predict what they should see. They assume that cells behave like simple, predictable machines where the past doesn't matter (Markovian).

• The Problem: Cell division isn't simple. It's messy, irregular, and depends on history. Trying to write a math equation for this is like trying to predict the exact path of a leaf blowing in a hurricane using a ruler. It's too complex; the math breaks.

The New Way (The "Video Game" Approach):
The authors realized that while writing the math equation is impossible, simulating the process is easy.

• The Analogy: Think of a video game. You can't write a single equation that predicts exactly where every pixel will be in a complex 3D world. But you can run the game. You can press "start," let the game run, and see what happens.
• The Innovation: The authors built a "Video Game Simulator" for cells. They ran millions of simulations with different settings (e.g., "What if the factory makes toys fast?" vs. "What if it makes them slow?").

The Secret Weapon: The "Neural Network Translator"

Here is the tricky part: They have the simulator, but they need to go backward. They have the real data (the snapshot of the factory), and they need to find the settings that created it.

Usually, you need a math formula to do this. Since they don't have one, they used a Neural Network (a type of AI) to act as a translator.

1. Training the AI: They fed the AI millions of pairs of data:
   • Input: "Here are the factory settings (e.g., fast production)."
   • Output: "Here is the result of the simulation (the snapshot of toys)."
2. The Learning: The AI learned to recognize the pattern. It learned, "Oh, when I see this specific pattern of toy counts, it usually comes from these specific settings."
3. The Result: The AI became a "Likelihood Machine." It could look at a real snapshot and say, "I'm 99% sure this came from a factory running at 5% speed, even though it looks like it's running at 100%."

The Real Discovery: The "Sleeping Giant"

They applied this new method to yeast cells (a type of fungus) under stress (starvation). They were looking at a gene called glc3, which helps the yeast store energy.

What they expected:
The yeast was starving (high stress). The fluorescence (glow) was super bright. They thought, "The cells must be frantically working to make energy proteins!"

What they actually found:
Using their "Video Game + AI" method, they discovered the opposite.

• The cells were actually mostly asleep (the gene was inactive 95% of the time).
• However, every once in a while, a cell would wake up for a split second, make a huge burst of proteins, and go back to sleep.
• Because the proteins last a long time and get passed down to children, the "heirlooms" piled up.

The Analogy:
Imagine a house that looks like a mess of toys everywhere.

• Naive View: "The kid must be playing with toys right now!"
• Real View: "The kid hasn't played in weeks. But every month, they get a new box of toys from their parents, and they never throw the old ones away. The mess is just a pile of old boxes."

Why This Matters

This paper is a game-changer because it stops scientists from making the "Naive View" mistake.

1. It fixes the math: It admits that cell division is messy and history-dependent, so it stops trying to force simple math on it.
2. It uses simulation: Instead of solving impossible equations, it uses the power of computers to simulate reality and then uses AI to interpret the results.
3. It reveals the truth: In the yeast example, it showed that the cells aren't constantly stressed; they are using a "bet-hedging" strategy. They stay mostly inactive to save energy, only waking up briefly to stockpile supplies, which they then pass down to their children.

In short: The authors built a time-machine simulator and an AI detective to separate "newly made" proteins from "inherited" ones, revealing that cells are often much more efficient and lazy than we thought.

Technical Summary

1. Problem Statement

The central challenge addressed is the inference of protein production kinetics (specifically transcriptional switching rates and production rates) in dividing cells using snapshot data (e.g., flow cytometry).

• The Limitation of Standard Models: Traditional approaches model protein dynamics using memoryless (Markovian) birth-death processes, assuming protein reduction occurs primarily through degradation. They rely on analytical likelihoods derived from the Chemical Master Equation (CME).
• The Biological Reality: In many biological systems, proteins have half-lives significantly longer than the cell cycle. Consequently, protein reduction occurs mainly through partitioning during cell division rather than degradation.
• The Non-Markovian Barrier: Because protein counts in a daughter cell depend on the specific division history of the mother cell (and the timing of previous divisions), the system becomes non-Markovian. Standard CMEs and likelihood functions cannot be formulated analytically for these general settings, rendering traditional maximum likelihood estimation and Bayesian inference inapplicable.
• The Data Challenge: Experimental data (fluorescence) often reflects a mixture of newly produced proteins and those inherited from previous generations, making it difficult to distinguish between a gene that is constitutively active and one that is transiently activated but whose products persist.

2. Methodology: Simulation-Based Inference (SBI) with Conditional Normalizing Flows

The authors propose a framework to perform inference without an explicit analytical likelihood function, relying instead on forward simulations.

A. The Core Approach: Likelihood Approximation

Instead of approximating the posterior directly (which can lead to overconfident estimates without validation), the authors approximate the likelihood function $p(\text{data}|\theta)$ using neural networks.

• Simulation-Based Inference (SBI): They generate synthetic datasets by simulating the biological process for various parameter sets $\theta$.
• Conditional Normalizing Flows (CNFs): They employ CNFs, a class of neural networks designed to learn complex probability distributions.
  • The network learns a bijective, invertible transformation $f_\phi$ that maps a simple base distribution (e.g., Gaussian) to the target conditional distribution $p(\text{data}|\theta)$.
  • Training: The network is trained by minimizing the negative log-likelihood of simulated data pairs $(\theta_s, y_s)$. The loss function is:
    $$L(\phi) = -\frac{1}{S} \sum_{s=1}^S \log p_{NF}(y_s | \theta_s, \phi)$$
  • Dynamic Resampling: To prevent overfitting, the training process involves dynamically resampling parameters and regenerating synthetic data at each epoch.

B. The Three-Stage Model Hierarchy

The authors validate their method using a hierarchy of increasingly complex models:

1. Model 1 (Deterministic Division): Cells divide at fixed intervals $T$. Protein production is constant ($\beta$). Partitioning is binomial.
   • Validation: An exact analytical solution for the steady-state distribution (an incomplete Gamma function) exists. The neural network successfully approximates this known solution.
2. Model 2 (Stochastic Division): Division times follow a Gamma distribution (capturing cell cycle variability).
   • Challenge: The likelihood becomes analytically intractable. The method is validated by comparing the learned likelihood against simulation histograms.
3. Model 3 (Realistic Biological Complexity):
   • Two-State Regulation: Genes switch between active ($\lambda_{act}$) and inactive ($\lambda_{ina}$) states.
   • Asymmetric Division: Models budding yeast ($S. cerevisiae$) where daughter cells inherit proteins based on volume fraction (Beta distribution), not just 50/50.
   • Indirect Measurement: Incorporates autofluorescence and a calibration factor to map protein counts to observed fluorescence intensity.
   • Parameters: $\theta = \{\beta, \sigma, \lambda_{act}, \lambda_{ina}\}$.

C. Inference Pipeline

1. Training: Train the CNF on synthetic data generated from the simulator.
2. Posterior Estimation: Use Bayes' theorem with the learned likelihood:
   $$p(\theta | \{y_n\}) \propto \left( \prod_{n=1}^N p_{NF}(y_n | \theta, \phi^*) \right) p(\theta)$$
3. Sampling: Use Markov Chain Monte Carlo (MCMC) to sample from the posterior distribution of the parameters given real experimental data.

3. Key Results

A. Validation on Synthetic Data

• Model 1: The neural network accurately recovered the exact analytical likelihood, demonstrating that the CNF can learn the correct functional form of the distribution.
• Model 2 & 3: The method successfully inferred parameters ($\beta, \sigma, \lambda_{act}, \lambda_{ina}$) from synthetic data. The posterior distributions tightened as the dataset size increased, and the method could distinguish between different kinetic regimes (e.g., high vs. low activation rates).

B. Application to S. cerevisiae Flow Cytometry Data

The method was applied to experimental data monitoring the glc3 gene (involved in glycogen synthesis) under nutrient-limiting conditions (high stress) vs. nutrient-rich conditions (low stress).

• Observation: High-stress conditions showed significantly higher fluorescence intensity than low-stress conditions.
• Naive Interpretation: One might assume cells are constitutively active under stress.
• Inferred Kinetics: The SBI framework revealed a counter-intuitive truth:
  • Cells under high stress are rarely active (only 5% of the time) compared to low stress (2%).
  • However, when the gene does activate, the produced proteins are long-lived.
  • Because proteins are inherited across multiple cell divisions, the accumulation of inherited proteins creates a high fluorescence signal, even though the gene is only transiently active.
• Conclusion: The high fluorescence is a result of protein inheritance, not continuous production.

4. Significance and Contributions

1. Overcoming Non-Markovian Barriers: The paper provides a general solution for inferring kinetics in systems where cell division history matters, a scenario where standard master equations fail.
2. Likelihood-Based SBI: By focusing on approximating the likelihood (rather than just the posterior) using Normalizing Flows, the authors maintain the statistical rigor of likelihood-based inference. This allows for better validation and avoids the "overconfidence" issues seen in posterior-only approximation methods.
3. Biological Insight: The study fundamentally changes the interpretation of gene expression data in dividing cells. It demonstrates that high protein levels do not necessarily imply high transcriptional activity; they can be an artifact of slow degradation and cell division inheritance.
4. Methodological Flexibility: The framework is modular. It can incorporate complex biological features (asymmetric division, state switching, measurement noise) that are difficult to model analytically but easy to simulate.
5. Practical Utility: The authors provide a scalable workflow (implemented in Python/GPU) that bridges the gap between complex biological simulators and rigorous statistical inference, applicable to proteomics, gene regulation, and cell cycle studies.

In summary, this work establishes a robust computational framework to disentangle production from inheritance in single-cell data, revealing that the "memory" of cell division can mask the true, transient nature of gene activation.