Size-structured populations with growth fluctuations: Feynman--Kac formula and decoupling

This paper generalizes the decoupling of internal fluctuating variables from cell size in size-structured populations by deriving conditions for lineage and ensemble decoupling, connecting these results to the Feynman-Kac formula to transform size dynamics into growth-homogeneous processes via random time changes and exponential tilting.

The Gist

Imagine a bustling city of single-celled organisms, like bacteria, growing and dividing in a petri dish. This paper is about understanding the hidden rules that govern how these cells grow, how they divide, and how their internal "personalities" (like gene activity) interact with their physical size.

Here is the story of the paper, broken down into simple concepts and everyday analogies.

1. The Two Characters: The "Engine" and the "Chassis"

Every cell in this model has two main features:

• The Chassis (Size): How big the cell is.
• The Engine (Growth Phenotype): How fast the cell's internal machinery is working. This is like a fluctuating engine RPM, determined by things like gene expression levels. Sometimes the engine sputters; sometimes it roars.

The big question the authors ask is: Does the size of the car (chassis) tell you anything about how fast the engine is running?

In many previous models, scientists assumed these two were tangled together. If a cell is big, maybe it's because its engine has been running fast for a long time. But this paper investigates a fascinating possibility: Decoupling. This is when the engine and the chassis become independent. Knowing the size tells you nothing about the engine, and vice versa.

2. The Two Viewpoints: The "Family Tree" vs. The "Crowd"

To understand the population, you have to look at it from two different angles, which often give different answers:

• The Lineage View (The Family Tree): Imagine following one specific cell and its direct descendants down a single family tree. You watch one line of cells grow, divide, and pass traits down.
• The Population View (The Crowd): Imagine taking a snapshot of the entire petri dish at one moment. You see millions of cells.

The Twist: In a growing population, the "Crowd" is biased. Fast-growing cells reproduce more often, so they make up a larger percentage of the crowd than they do in a single family tree. It's like a raffle where fast-growing cells buy more tickets. If you only look at the family tree, you might miss the fact that the fast growers are dominating the room.

3. The Magic Trick: "Decoupling"

The authors discovered that under certain biological conditions, the "Engine" and the "Chassis" stop influencing each other.

• The Analogy: Imagine a factory where workers (cells) build boxes (biomass).
  • Normal Mode: The speed at which a worker builds a box depends on how big the box already is. This creates a messy, tangled relationship.
  • Decoupled Mode: The worker has a personal speed setting (the Engine) that changes randomly but is blind to the box size. The box size just grows at that speed.
  • The Result: If the worker's speed setting doesn't change when they split the box in half (division), then the speed setting and the box size become independent. You can study the speed settings without worrying about the box sizes, and vice versa.

The paper proves exactly when this magic happens. It happens if the "rules of division" are simple enough that the internal speed setting doesn't get "reset" or "distorted" when the cell splits.

4. The Mathematical Magic Wand: The Feynman-Kac Formula

This is the most technical part, but here is the simple version.

The authors use a mathematical tool called the Feynman-Kac formula. Think of this as a time-traveling translator.

• The Problem: It's very hard to calculate what the whole "Crowd" looks like because of the bias (the fast growers).
• The Solution: The formula allows you to take data from a single "Family Tree" (which is easier to simulate or measure) and mathematically "tilt" or "weight" it to predict the "Crowd."
• The Analogy: Imagine you are trying to guess the average height of everyone in a stadium, but you can only interview people on one specific row (the lineage).
  • The Feynman-Kac formula gives you a formula to say: "If I take the people on this row, and I give extra weight to the ones who are tall and fast-growing, I can accurately predict the average height of the entire stadium."

This "tilting" is essentially an importance sampling technique. It tells you how to look at your single-cell data so that it looks like the whole population.

5. Why This Matters

This isn't just abstract math; it has real-world uses:

1. Better Experiments: Scientists often use "Mother Machines" (microfluidic devices) to watch single cells grow. This paper tells them exactly how to translate those single-cell observations into accurate predictions about the whole bacterial culture.
2. Gene Expression Noise: It helps explain how random fluctuations in genes (noise) affect a population. Sometimes the noise cancels out; sometimes it amplifies. This model helps predict which happens.
3. Simpler Simulations: If the "decoupling" happens, scientists can run much simpler computer simulations. They don't need to track the complex relationship between size and speed; they can just track them separately and combine the results.

Summary

The paper is a guidebook for understanding how individual cells grow and divide. It reveals that under the right conditions, a cell's internal speed and its physical size become independent strangers. When this happens, we can use a clever mathematical "translator" (Feynman-Kac) to take simple data from a single family line and accurately predict the behavior of the entire, complex population. It turns a messy, tangled problem into a clean, solvable one.

Technical Summary

1. Problem Statement

The paper addresses the challenge of modeling size-structured populations where individual cells exhibit phenotypic heterogeneity in their growth rates. Specifically, it investigates the relationship between two distinct statistical ensembles:

1. The Lineage Ensemble: The distribution of phenotypes observed along a single, continuous cell lineage (tracking one daughter cell per division).
2. The Population Ensemble: The snapshot distribution of phenotypes across the entire growing population at a given time.

Key Challenge: In growing populations, faster-growing cells are overrepresented due to selection bias. This creates a systematic difference between lineage and population distributions. Previous models often treated growth rates as fixed or decoupled from cell size. However, when growth rates ($\lambda$) fluctuate based on internal variables (e.g., gene expression $X$) and division rates ($\beta$) depend on both size and these internal variables, the joint distribution of size and phenotype becomes complex. The authors aim to determine:

• Under what conditions do cell size and growth phenotype decouple (become independent) in the lineage and/or population ensembles?
• How can population statistics be rigorously derived from lineage statistics using Feynman–Kac formulas?
• What is the physical interpretation of these statistical relationships when decoupling does not occur?

2. Methodology

The authors develop a general mathematical framework for size-regulated growth with fluctuating phenotypes.

A. Model Formulation

The model tracks a cell's state via:

• Growth Phenotype ($X(t) \in \mathbb{R}^d$): Evolves according to a Markov process with generator $L$ (e.g., Ornstein-Uhlenbeck process) independent of size between divisions.
• Size Phenotype ($Y(t) = (y_c, y_b)$): Current log-size ($y_c$) and birth log-size ($y_b$). Size grows exponentially at rate $\lambda(X(t))$.
• Division: Occurs at a rate $\beta(X, Y)$. Upon division, phenotypes are sampled from a kernel $h$.

The dynamics are described by Partial Differential Equations (PDEs) for the joint density $\rho(x, y)$:

• Lineage PDE: Includes advection (growth) and a loss term due to division.
• Population PDE: Similar to the lineage PDE but includes a source term (division produces two cells) and a dilution term to account for population expansion.

B. Structural Conditions

The authors define two critical conditions to analyze decoupling:

• Condition 1 (Division Hazard Factorization): $\beta(x, y) = \lambda(x)\phi(y)$. This implies the division hazard per unit of added size depends only on size, not the internal state.
• Condition 2 (Inheritance Factorization): $h(x, y_b | x', y'_c) = r(x|x')u(y_b|y'_c)$. This implies independent inheritance noise for the internal phenotype versus size partitioning.

C. Operator Splitting and Time-Change

The core mathematical tool is Operator Splitting. The authors reformulate the steady-state equations as fixed-point problems involving linear operators.

• They introduce a random time-change $T(t) = \int_0^t \lambda(X(s)) ds$. This transforms the size dynamics into a process with a constant growth rate, effectively separating the time-scale of size accumulation from the internal state evolution.
• They utilize the Feynman–Kac formula, which relates the expectation of a functional of a stochastic process to the solution of a parabolic PDE (or an eigenvalue problem).

3. Key Contributions and Results

A. Decoupling Regimes

The paper rigorously defines and proves conditions under which the joint distribution $\rho(x, y)$ factorizes into $\nu(x)w(y)$ (growth phenotype $\times$ size distribution).

1. Strong Decoupling (SD):

   • Conditions: Condition 1 & 2 hold, and the division kernel preserves the internal state ($r(x|x') = \delta(x-x')$).
   • Result: Decoupling occurs in both lineage and population ensembles.
   • Mechanism: The population growth rate $\Lambda$ is the dominant eigenvalue of the tilted generator $\hat{L}_\lambda = L + \lambda$. The population density $\nu_p$ is the corresponding eigenfunction.

2. Weak Decoupling in Lineage (WD):

   • Conditions: Condition 1 & 2 hold, but $X$ changes at division ($r \neq \delta$). The iteration $\nu_{k+1} = R_\lambda \nu_k$ converges to a fixed point in $\ker(L)$.
   • Result: Decoupling holds only for the lineage distribution. The population distribution generally remains coupled because the population ensemble introduces a "dilution" bias that breaks the symmetry required for factorization.

3. Weak Decoupling in Population (WDp):

   • A mathematically possible but biologically rare case where decoupling holds for the population but not the lineage.

B. Lineage Representation of Population Statistics (Feynman–Kac)

The authors derive a "Many-to-One" formula connecting the two ensembles. For Strong Decoupling, the population expectation of a function $f(X)$ is given by a tilted expectation over lineage paths:

$$E_p[f(X(t))] = \lim_{t \to \infty} \frac{E_\ell^{\text{path}}\left[ f(X(t)) e^{\int_0^t \lambda(X(s)) ds} \right]}{E_\ell^{\text{path}}\left[ e^{\int_0^t \lambda(X(s)) ds} \right]}$$

• Interpretation: The population distribution is an exponentially tilted version of the lineage distribution. The weighting factor $e^{\int \lambda ds}$ corresponds to the total biomass accumulated along the lineage.
• Numerical Implication: This allows estimating population statistics from lineage data (e.g., Mother Machine experiments) using importance sampling, though it requires exponentially many lineages for convergence due to the variance of the weight.

C. General Case: Mass-Weighted Phenotype Distribution

When decoupling does not occur (e.g., $X$ and size are correlated), the authors show that the Feynman–Kac formula still holds but with a different interpretation.

• The term $e^{\int \lambda ds}$ can be decomposed into $e^{y_c(t)} 2^{K(t)}$ (current mass $\times$ number of descendants).
• Consequently, the tilted expectation represents the mass-weighted average of the phenotype in the population, rather than the count-weighted average.
• This provides a generalized interpretation: Population statistics are always related to lineage statistics via a size-biasing mechanism, even without strict decoupling.

4. Significance and Applications

• Unification of Models: The framework generalizes previous models (adder, sizer, random growth rate) and clarifies when they are valid approximations.
• Experimental Design: It provides a rigorous theoretical basis for inferring population-level selection pressures from single-cell lineage data, a common practice in modern biophysics.
• Algorithmic Efficiency: The results suggest new Monte Carlo simulation strategies. By using the decoupled dynamics or the tilted generator as a proposal distribution, one can reduce the variance in estimating population growth rates and phenotype distributions.
• Biological Insight: The paper clarifies that "size regulation" mechanisms (like adder/sizer) can effectively "hide" the complexity of fluctuating growth rates, allowing size distributions to stabilize even when internal states are highly dynamic, provided specific factorization conditions are met.

5. Conclusion

Levien et al. successfully bridge the gap between stochastic single-cell dynamics and deterministic population-level descriptions. By leveraging the Feynman–Kac formula and operator splitting, they establish precise conditions for the decoupling of size and phenotype. Their work demonstrates that population statistics can be viewed as importance-sampled lineage statistics, offering both a theoretical understanding of selection bias and practical tools for analyzing high-resolution single-cell data.