Biology from First Principles: How Often Should You Measure?

In my previous post, I explored what questions matter in biology. Now I want to tackle a more specific one: once we know what to measure, how do we figure out how often to measure it? This seemingly simple question has major implications for experimental design. Of course, figuring out what exactly we want to measure is also challenging, and we will tackle that in future. But for now, let's assume we know our target and focus on the timing question. Surprisingly, there's still no clear quantitative framework for determining this. In this article I'll be improvising using physics to tackle this problem.

First of all, I apologize for awkward Latex throughout the text since it seems like Substack does not support inline Latex… Let's introduce some notation. In this article let's look at a cell as a single agent which is navigating a latent transcriptional space denoted as

Cell density at each point x and time t could be denoted as c(x,t). We also know that a cell is a stochastic system. For example a hematopoietic stem cell can give rise to many different blood cell types, so they effectively have a directed but stochastic trajectory. This framework is commonly used in modelling of physical systems.

We can think of cells experiencing two types of movement through gene expression space. First, their developmental programs create systematic drift toward specific cell fates, represented by

Here, J represents the velocity field of cellular development and it tells us which direction and how fast cells want to move at each point in gene expression space.

Second, inherent biological noise causes random fluctuations around this directed path, modeled by the diffusion term

This allows us to arrive to a version of a stochastic equation called Fokker-Plank equation:

To build intuition, let's first consider the simpler case where cells have no developmental program, they just randomly wander through gene expression space. This means we can ignore the drift term J for now and focus purely on the diffusion component. Think of this as studying how much cells naturally spread out over time due to inherent biological noise before we add the complexity of directed cellular programs.

So basically we are asking: If we place cells at a specific gene expression state and let them sit, how much will they spread out purely due to stochastic fluctuations in transcription and translation?

For a purely diffusive step between two snapshots spaced by Δt the mean-square displacement is

The formula above comes from the fundamental property of diffusion processes where the variance scales linearly with time (think of a random walk where each step is independent).

The measurement that we are going to perform on the system adds and independent localisation error which becomes:

Note: η² is instrumental noise (library + prep counting), and it does not vanish with longer Δt, whereas 2DΔt grows with Δt

The optimal sampling time balances two competing effects. Sample too quickly (DΔt < η²) and measurement noise dominates over biological signal. Sample too slowly (DΔt > η²) and you waste experimental effort without gaining information. The sweet spot occurs when biological variance equals measurement variance: DΔt = η², giving us

More formal exploration of this you can find here.

Now let's add back the developmental program (drift term). The full equation is complex because J(x) can have any shape, cells might be attracted to some gene expression states and repelled from others. To move forward, we add a trick called linearization.

We focus on small movements around a stable point x₀ where J(x₀) = 0 (meaning that cells want to stay there). Near this point, we can approximate the complex drift as a simple spring force (doing Taylor expansion). So basically when cells drift away from x₀, the developmental program pulls them back proportional to how far they moved.

Mathematically, this means J(x) ≈ -A(x - x₀) , where A is the spring constant of a developmental program. This transforms our complex PDE into the much more tractable Ornstein-Uhlenbeck process:

We basically get Ornstein-Uhlenbeck(OU) process, which is basically the brownian motion tethered by a spring of strength A. if we take the expectation of this equation, we will find the relaxation time is τ = 1/A:

The noise has zero mean, and we are able to eliminate the randomness term. The key insight is that while individual cells are constantly jittering around due to noise, the average behavior of the cell population follows a predictable exponential decay back to the target state. The time constant τ tells us how fast this return happens, and you may also remember from your differential equations class that one τ leads to deviation drop of about 37% of initial size.

Outcome:

1. If you sample slower than τ you only see the after state, the drift shape is lost

2. If you sample faster than η²/D you basically waste your effort since the measurement error will dominate.

So again, we have two competing constraints: sample too slowly and miss the biological dynamics (lose the signal), sample too quickly and drown in measurement noise (waste resources). The optimal timing balances these trade-offs.

NIH-3T3 fibroblasts are the canonical model for growth-factor signaling and IEG kinetics. Literature shows that serum drives the first wave of IEGs within minutes, with c-fos transcription peaking by 15 minutes and returning to basal levels within 60 minutes. To estimate realistic velocity parameters, we turn to RNA velocity studies, which show that cellular differentiation occurs on timescales of hours. Literature showed that the characteristic extrapolation time (i.e., the time over which RNA velocity predictions are accurate) was about 2.5 to 3.8 hours. Taking the middle of this range (3.15h) gives v = 0.317 h⁻¹ (1/3.15h) and leads to D=⟨v²⟩/2=0.05025. Let's also take technical noise of 5% from the spike-in and variance-stabilisation studies of 10x UMI data. Plugging those numbers into the noise balance formula we get an optimal sampling interval of about 3 mins.

We must also sample faster than the fastest deterministic change. For this, we need to look at our characteristic time constant. Literature reports canonical spliced c-fos mRNA (lacking intron 3) with a half-life of about 15 minutes.

Taking the stricter condition, we get a requirement of one snapshot every 3 minutes. The current literature considers high-temporal resolution to sample a cell culture every 10 minutes, but our calculations suggest that you would need to sample even faster.

This model makes several simplifying assumptions like linearization around equilibrium points and ignoring cell-cell interactions, among others. These would break down in more complex systems. However, even with these limitations, this quantitative framework provides more principled guidance than the qualitative, intuition-based approaches typically used for experiments.

This framework gives experimentalists a principled way to choose sampling intervals rather than relying on intuition or convention. My goal here more broadly is to demonstrate how physics can provide quantitative answers to seemingly qualitative biological questions. Future posts will explore how these same principles apply to other areas of biology.