2C: Active cell layers, from Vicsek to epithelia

Elective project, round 1 (P2). Report due Sun 25 April, defense Tue 27 April

The question

Epithelial cells, the cells that line our organs, form dense layers in which they push, crawl and align with their neighbours. Physicists describe such layers as active matter, the same class as the flocks of the Vicsek model. Images of real cell layers often seem to show regions of aligned cells. But a pattern that looks ordered to the eye can also arise by chance. Your question is:

Do the cells in an epithelial monolayer align with each other, beyond what chance would give?

You will first map the order–disorder transition in the Vicsek model, where you know the answer, and use it to build and test your tools. Then you apply the same tools to quantitative phase images of MDCK epithelial monolayers from our group.

Material

Milestones

When What you show
M1 Tue 30 Mar (W13) Vicsek order parameter against noise for at least two densities, and the MDCK images segmented
M2 Tue 13 Apr (W15) Validated core figure: nematic order of the cells against window size, compared with a null model, with the method first tested on simulated data where the answer is known
M3 Sun 25 Apr Report and LLM log on Canvas
M4 Tue 27 Apr (W17) Oral defense

What the project should contain

1. The flocking transition

  • Run the Vicsek model for a grid of noise amplitudes \(\eta\) and densities \(\rho = Nr^2/L^2\). Plot the polar order parameter \(\varphi = \left|\frac1N\sum_j e^{i\theta_j}\right|\) in the steady state against \(\eta\) for each density.
  • How do you know the simulation has reached a steady state? How large are the fluctuations of \(\varphi\)?
  • Finite-size check. For fully random orientations, \(\varphi\) is not zero but of order \(1/\sqrt N\). Show this, and use it to decide when a measured \(\varphi\) means order.

2. Cell shape and orientation in the monolayer

  • Segment the cells with the notebook. Check the segmentation by eye on a part of the image, and estimate how often it splits or merges cells.
  • For each cell, compute the area, the aspect ratio and the orientation of the long axis from the second moments of its shape. Cell orientation is a nematic quantity: a cell pointing at \(\theta\) is the same as one pointing at \(\theta + \pi\).
  • Plot the distributions of area and aspect ratio, and how they change with cell density.

3. Is there nematic order?

  • Define the nematic order parameter in a window of radius \(R\), \(S(R) = \left|\langle e^{2i\theta_j}\rangle_{j\in R}\right|\), averaged over many windows, and plot it against \(R\).
  • Null model. Keep the positions and shapes of the cells, but shuffle the orientations randomly between cells. Repeat many times to get the distribution of \(S(R)\) for a layer with no correlations. With \(n\) cells in a window, a random layer gives \(S\approx\sqrt{\pi/(4n)}\). Is the measured \(S(R)\) outside the null distribution? At which \(R\)?
  • Test on known data. Before you trust part 3 on cells, apply exactly the same analysis to a Vicsek simulation (use \(e^{2i\theta}\) of the particle directions) at low and at high noise. The method must find order where there is order, and none where there is none.

4. Discussion

What does your result say about whether an epithelial layer behaves like a Vicsek flock? What would you need to measure to tell them apart: polar or nematic order, motion or shape?

Working with an LLM on this project

Image analysis invites a lot of LLM-written code, and it is easy to get impressive-looking orientation maps. Insist on the null model: every statistic you report must be compared with what the same analysis gives for random data. An LLM will rarely suggest that on its own.

The oral defense

About 15 min per student. Typical questions:

  • Explain the Vicsek model. What are the control parameters, and what is the order parameter?
  • Why is the order parameter of a random system not zero? How does it scale with the number of particles?
  • What is the difference between polar and nematic order? Which one is right for cell shapes, and why?
  • Explain your null model. What does it keep, and what does it destroy?
  • What does a quantitative phase image measure, and how is that turned into a cell outline?
  • What did the LLM get wrong?