How Cells Really Grow: A New Way to Decode Biological Individuality

5% measurement noise can make cell growth look 300% more variable than it really is — unless you use this new method.

That’s not a flaw in the microscope. It’s a flaw in how we’ve been analyzing biological data for decades.

When scientists track how individual cells grow over time — recording their size, division, and metabolism — they’re not just watching life unfold. They’re trying to reverse-engineer the hidden rules that govern it. For years, the standard approach has been simple: fit a curve to each cell’s trajectory, extract its growth rate, then treat the resulting spread as biological truth. But here’s the catch: some of that spread isn’t biology. It’s noise. Measurement error. Numerical instability. And until now, we had no reliable way to tell the difference.

Now, a team at the University of Colorado has developed a framework that does exactly that. Their method, built on a technique called Weak-form Estimation of Nonlinear Dynamics (WENDy), doesn’t just fit curves — it separates signal from artifact, distinguishing real physiological differences between cells from distortions introduced by imperfect data. In doing so, it transforms our ability to understand not just how cells grow, but how populations emerge from individual variation.

This isn’t incremental progress. It’s a shift in perspective — from treating cells as interchangeable units to recognizing them as individuals whose differences matter.

The Science

The problem starts with a deceptively simple question: How does a cell grow?

At first glance, the answer might seem obvious. Maybe it grows exponentially — doubling in size at a constant rate. Or perhaps it follows a logistic curve, slowing as it approaches a maximum size. These are classic models, taught in textbooks and used in thousands of papers. But modern experiments reveal something deeper: no two cells follow the same path. Even genetically identical cells in the same environment grow at different rates, reach different sizes, and divide at different times.

This individuality isn’t random noise. It’s meaningful. Heterogeneity in growth rates can drive population-level phenomena like multi-modal size distributions, differential survival under stress, and even the emergence of drug-resistant subpopulations in cancer or bacterial infections (Kiviet et al., 2014). To predict these outcomes, we need to know not just the average behavior, but the full distribution of individual traits.

Yet traditional methods fall short. Fitting nonlinear models to noisy trajectories is unstable. Small errors in measurement lead to large errors in estimated parameters. Worse, when researchers compare multiple candidate growth laws — say, exponential versus logistic — they often rely on repeated simulations and optimization, which are computationally expensive and statistically fragile.

Enter WENDy. Developed originally for parameter estimation in dynamical systems, WENDy avoids numerical differentiation entirely by working in the “weak form” — integrating the governing equation against smooth test functions. This turns the problem into a regression task where derivatives are moved from the noisy data to known, analytically differentiable functions. As a result, the method is far more robust to noise than conventional approaches.

But the authors go further. They embed WENDy within a hierarchical statistical model — a random-effects framework — that treats individual parameters as draws from an underlying population distribution. After estimating parameters for each cell using WENDy, they apply empirical Bayes shrinkage to correct for overdispersion caused by measurement noise. The idea is simple: if all cells were identical, any observed variation would be pure noise. If they’re highly diverse, the spread should remain after correction. By pooling information across the population, the method estimates the true level of biological heterogeneity.

Finally, they use this corrected distribution to upscale predictions to the population level, linking individual trajectories to structured population models — hyperbolic partial differential equations that describe how size distributions evolve over time.

The full pipeline looks like this: for each of several candidate growth laws (e.g., exponential, logistic, von Bertalanffy), apply WENDy in parallel to every cell’s trajectory; compute a weak-form version of the Bayesian Information Criterion (BIC) summed across all individuals; select the model with the best score; then map the fitted linear-in-parameters weights back to biologically interpretable quantities (like growth rate $r$ and carrying capacity $S_{\max}$); finally, apply empirical Bayes shrinkage to recover the true parameter distribution $\mathcal{P}$.

What They Found

The results are striking — especially when noise enters the picture.

In synthetic tests, the method correctly identified the true growth law (e.g., metabolic von Bertalanffy) in over 95% of cases, even with 5% additive noise — a realistic level for live-cell imaging. More importantly, it recovered the true distribution of growth parameters with high fidelity.

Consider one key finding: without correction, the estimated variance of growth rates inflated dramatically with noise. At just 5% measurement noise, the raw WENDy estimates suggested three times more variability than actually existed

Raw vs. Shrunken Variance Estimates Across Noise Levels

Label	Value
0%	0
2%	0
5%	0
10%	0

True biological variance (τ²) = 0.01. Even moderate noise causes raw estimates to diverge sharply.

. That kind of error would mislead any attempt to link cellular physiology to population dynamics.

But the empirical Bayes step fixed it. By shrinking extreme values toward the population mean — a process akin to regressing batting averages in baseball — the method recovered the true underlying spread. The correction was so effective that even at high noise levels, the estimated biological variance stayed within 20% of ground truth.

When applied to real data — single-cell E. coli trajectories from Tanouchi et al. (2015) grown in three different media — the method revealed something subtle but profound: the degree of growth heterogeneity depends on environment.

In glucose-casamino acids (rich medium), cells grew fast and uniformly — coefficient of variation (CV) in growth rate was just 22%. In glycerol (poorer carbon source), growth was slower but far more variable — CV jumped to 41%. Alanine fell in between at 34%

Biological Coefficient of Variation in E. coli Growth Rates

Label	Value
Alanine	34 %
Glycerol	41 %
Glucose-Casamino Acid	22 %

Higher BCV indicates greater individuality in growth response. Glycerol-grown cells show most heterogeneity.

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These aren’t just numbers. They suggest that environmental stress amplifies individuality — perhaps because small differences in enzyme efficiency or metabolic regulation become magnified when resources are scarce. This aligns with broader ecological principles: variability buffers populations against uncertainty. But now, we can quantify it directly from trajectory data.

The method also worked on sparse, irregular data — like historical loblolly pine growth records — successfully identifying linear von Bertalanffy growth despite missing time points and low sampling frequency

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And in artificial cohort mixture experiments, where subpopulations followed distinct growth programs, the shrunken parameter estimates enabled accurate reconstruction of population density evolution — outperforming models based on mean parameters alone

.

Why This Changes Things

For decades, population biology has wrestled with a paradox: we study collectives, but our data come from individuals. Bridging that gap requires assumptions — often untested — about how individual variation translates into group behavior.

This work offers a solution grounded in both physics and statistics. It treats growth as a dynamical process governed by differential equations, while acknowledging that parameters vary across individuals according to a probability distribution. By combining weak-form inference with hierarchical modeling, it provides a unified framework for learning both the law and the luck — the deterministic rules and the stochastic diversity.

That matters because heterogeneity isn’t just a detail. In cancer, a small subpopulation of slow-growing cells can survive chemotherapy and seed relapse. In microbiology, phenotypic variation enables bet-hedging strategies that ensure survival in fluctuating environments. In ecology, trait variation alters species interactions and ecosystem stability.

Previous attempts to model these effects relied on assumed distributions — often Gaussian — with variances tuned by hand or inferred indirectly from bulk measurements. Now, we can measure the distribution directly, while correcting for observational error.

Moreover, the method enables true multiscale modeling. Once you have the distribution of individual growth laws, you can simulate how the entire population’s size structure evolves — predicting not just average biomass, but the full shape of the distribution, including tails and modes.

This has practical implications. In biotechnology, optimizing fermentation processes requires understanding not just mean productivity, but the spread of metabolic states. In medicine, predicting tumor growth may depend on identifying rare, fast-dividing clones. In conservation, managing fish stocks means accounting for size-dependent mortality across a heterogeneous population.

Even beyond biology, the framework applies wherever individual trajectories shape collective outcomes: neuron firing patterns in the brain, mobility patterns in cities, or career trajectories in labor markets.

What's Next

No method is perfect. The current implementation assumes that growth laws are known in advance — selected from a finite set of candidates. While powerful, this limits discovery of entirely new functional forms. Future versions could integrate with symbolic regression tools like SINDy to search open-ended libraries of possible models.

Another limitation: the assumption that parameters are independent across generations. In reality, daughter cells inherit physiological states from mothers. Extending the random-effects model to include maternal effects — perhaps via autoregressive priors — would bring the method closer to biological reality.

There’s also room to improve computational efficiency. While WENDy is embarrassingly parallelizable, the iteratively reweighted least-squares procedure can be slow for very long trajectories. Approximate inference techniques — variational Bayes or expectation propagation — might accelerate the shrinkage step.

Perhaps most exciting is the potential for experimental design. If we know how noise distorts parameter estimates, we can optimize sampling frequency, duration, and precision to maximize information gain. This could guide next-generation microfluidic devices that track cells for days with minimal phototoxicity.

Ultimately, this work reframes a fundamental question: When we observe variation in nature, how much of it is real?

The answer used to be guesswork. Now, it’s quantifiable.

As datasets grow — from high-throughput microscopy to longitudinal health records — the challenge won’t be collecting data. It will be interpreting it without being misled by noise. Methods like this one don’t just analyze data. They protect us from believing our own artifacts.

And in a world where personalized medicine, precision agriculture, and adaptive conservation all depend on understanding individuality within populations, that distinction isn’t academic. It’s essential.