Intermediate regime during finger formation: from instabilities to fingers

• Questions that we are interested in:– How do fingers break out of the edge?– Are there characteristic and dominant instability modes that lead to fingers?

• These would be wavelengths or modes that are unstable (will propagate over time)

• H ~ Σeλteikx

– How might these wavelengths relate to the resulting finger widths?

i

We are interested in the following transition:

The blue line indicates the boundary or interface between the inoculation and agarose.

• Define the boundary as points where the gradient in image intensity is large • Apply Fourier Transform to the boundary to find the Fourier modes that can describe the

shape of the boundary (I used the fft function in Matlab)• I then found the modes (excluding the first) that accounted for 90% (an arbitrary metric) of

the weights, as the modes that will sufficiently describe the shape.– The weights are found by taking the cumulative sum of abs(amplitude) for all modes, and then dividing

abs(amplitude) by this sum.– I then sorted the weights to find those that accounted for >= 90% of the cumulative sum.

The approach:

A demonstration of how the chosen modes (accounting for 90% of the sum of weights) sufficiently fits the boundary:

• To find the fast growing modes, we can look for modes with weights increasing over time– The colors correspond to different timepoints– The weights are taken relative to the cumulative sum of amplitudes at each timepoint

We are interested in growing modes so we zoom into this region with many growing modes (next slide)

I sampled the boundary at 10-min intervals

241.8134.3 (grew very fast)

151.1

100.8 (grew moderately fast)

93 (grew fast)

75.56 (emerged at later time points

Here I’m highlighting some modes that look like they are growing in weights over time

This looks promising, but is looking at the weights the right approach?

• Alternatively, we can look for modes with amplitudes increasing over time– On the y-axis, the amplitudes were normalized to the first timepoint that they emerged– The colors represent different instability wavelengths

Highlighting the top 10 modes (in microns), where amp(final)/amp(initial) > 15

9Aug data

40.30000031.815789109.90909152.56521734.54285748.36000033.58333341.68965527.47727350.375000

One can observe that the amplitudes are not always exponentially increasing with time, but the general trend does

An alternative plot to the same data. Here I did a 3D surf plot to visualize the amplitudes of the corresponding wavelengths over time

9Aug data

7Aug data

Highlighting the top 10 modes (in microns), where amp(final)/amp(initial) > 4

In the next few slides I will show results from 3 more experiments

151.12500093.000000100.750000134.333333109.909091120.900000172.71428663.63157967.166667

19Jan data 143.859375

164.410714127.87500041.10267942.62500038.36250039.68534544.264423230.175000191.812500115.087500

Highlighting the top 10 modes (in microns), where amp(final)/amp(initial) > 5.5

4Sept data

Highlighting the top 8 modes (in microns), where amp(final)/amp(initial) > 14

60.45000063.63157971.11764767.16666737.78125018.89062519.500000172.714286

Future work

• Comparison across more repeats and different starting cell densities to look for characteristic instability wavelengths that lead to finger formation– does it scale s.t. λ~ C1/3 ?

• Find the eigenvalue for the wavelengths with increasing amplitudes• Relate the instability wavelengths to the widths of resulting fingers

• Try to calculate the threshold cell density required for fingers to emerge

Quick conclusions

• We seem to see some wavelengths that emerge to rapidly increase their amplitudes over time, e.g. in the range between 30-50 um (on the scale of the small finger-like protrusions), and >100 um (close to the width of the bottom of the longer fingers that already formed by the end of the time considered)

Redefining “weight”• Before I defined “weight” as amplitude/(sum of amplitudes) for each

mode at that timestep• Now I will redefine “weight” as amplitude/(norm of amplitude vector), to

convert everything into unit vectors• The new plots are in the next few slides, and we can look for modes that

have non-exponential behavior (e.g. grew in weight then decreased)

Colormap represents time in multiples of 10 min

Example of weights by wavelengthFrom 9Aug data

Here I’m plotting the weight over time for each mode, on a log scale.The colormap represents the indices (not value) for each mode.

Here I plotted the time evolution of the normalized weight for the modes that experienced a large overall increase in weight

The very small modes (30-40um) may represent the small fluctuations that are forming on this cell-dense progressing front

109um mode

7Aug data

In the next few slides I will show results from 3 more experiments

Close to the 63um mode

Close to the 93um mode

19Jan data

The modes with value ~50+ um are representative of these small new fingers

This is approximately close to the 140+um mode

Do the fluctuations in the 52um mode indicate its instability (disappearing and reappearing) at earlier times before the emergence of the smaller fingers?

4Sept data

The modes with value ~60+ um are representative of these small new fingers

This is approximately close to the 172um mode