2D Diffusion & Mean Squared Displacement (MSD) Analysis

Introduction

Diffusion is the spontaneous net movement of particles from regions of higher concentration to regions of lower concentration driven by microscopic thermal fluctuations (Brownian motion). In cellular and molecular biophysics, single-particle tracking (SPT) microscopy enables researchers to trace coordinate trajectories of fluorescently tagged molecules, vesicles, and bacteria over time.

To extract the macroscopic diffusion coefficient ($D$) from discrete coordinate timeseries without continuous calculus approximations, we implement a discrete sliding-window Mean Squared Displacement (MSD) analysis.

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Theoretical Framework

Trajectory Coordinates

Consider a particle tracked across $N$ discrete time slices at uniform sampling intervals $\mathrm{\Delta }t=5.0\text{ seconds}$. The recorded trajectory is represented as a sequence of spatial coordinates:

$$\{(x_i,y_i){\}}_{i=1}^N,t_i=i\cdot \mathrm{\Delta }t$$

Mean Squared Displacement (MSD)

For an integer lag window $n\in \{1,2,\dots ,N-1\}$, the elapsed time lag is $\tau =n\cdot \mathrm{\Delta }t$. The time-averaged Mean Squared Displacement across all non-overlapping or overlapping intervals of duration $\tau$ is defined as:

$$\text{MSD}(\tau )=\frac{1}{N-n}\sum _{i=1}^{N-n}[(x_{i+n}-x_i{)}^2+(y_{i+n}-y_i{)}^2]$$

Extracting the Diffusion Coefficient ($D$)

In two-dimensional isotropic Brownian diffusion, Einstein's diffusion relation dictates:

$$\text{MSD}(\tau )=4D\tau$$

Plotting $\text{MSD}(\tau )$ against time lag $\tau$ yields a linear relationship whose slope $m$ satisfies:

$$m=4D⟹D=\frac{m}{4}$$

NOTE

The "Max Window" Statistical Anomaly:
As the lag window $n$ approaches $N$, the number of available independent displacement pairs $(N-n)$ diminishes rapidly. At $n=N-1$, only a single displacement pair is evaluated, causing statistical variance to diverge sharply. Standard biophysical protocol restricts linear regression to the initial $10\mathrm{\%}\text{--}25\mathrm{\%}$ of the MSD curve to ensure robust parameter estimation.

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Empirical Trajectory Dataset

The experimental trajectory consists of 65 sequential time slices ($\mathrm{\Delta }t=5\text{ s}$, total duration 325 seconds):

Time Slice	Time $t$ (s)	$X$ Coordinate	$Y$ Coordinate
1	5	88	15
2	10	87	16
3	15	85	18
4	20	81	21
5	25	80	21
10	50	67	25
20	100	50	11
30	150	46	20
40	200	48	21
50	250	50	19
60	300	40	14
65	325	44	18

(Full timeseries of 65 coordinate pairs archived in lab/projects/diffusion-coefficient/msd.py)

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Computational Methodology

The MSD pipeline is implemented in Python using NumPy and Pandas:

def calculate_msd(data, dt=5.0):
    N = len(data)
    lags = np.arange(1, N)
    msd_values = []

    for lag in lags:
        dx = data['x'].diff(periods=lag).dropna()
        dy = data['y'].diff(periods=lag).dropna()
        squared_displacement = dx**2 + dy**2
        msd_values.append(squared_displacement.mean())

    tau_values = lags * dt
    return tau_values, np.array(msd_values)

Linear regression over the initial 25% data cutoff ($\tau \le 80\text{ s}$) yields the slope $m$ and empirical 2D diffusion coefficient $D$:

$$D=\frac{\text{slope}(MSD\text{ vs }\tau )}{4}$$

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Interactive Marimo Notebook Pipeline (Future Integration)

The reactive computation script lab/projects/diffusion-coefficient/msd.py is configured with PEP 723 metadata:

# /// script
# requires-python = ">=3.13"
# dependencies = [
#     "marimo>=0.20.2",
#     "matplotlib==3.10.8",
#     "numpy==2.4.2",
#     "pandas==3.0.1",
# ]
# ///

Note: Per portfolio architecture specifications, full client-side WebAssembly (WASM) marimo runtime integration is scheduled for subsequent releases, with the analytical mathematical models and empirical timeseries preserved in native Markdown.

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References

1. Einstein, A. (1905). Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik, 322(8), pp. 549–560.
2. Qian, H., Sheetz, M.P. and Elson, E.L. (1991). Single particle tracking: analysis of diffusion and flow in two-dimensional systems. Biophysical Journal, 60(4), pp. 910–921.
3. Michalet, X. (2010). Mean square displacement analysis of single-particle trajectories with localization error: Brownian motion in an isotropic medium. Physical Review E, 82(4), 041914.