QBioS PhD Defense

Biofilms are 3D structures

complex surface-attached

Surface

Interface

Cells

Extracellular matrix formed of polysaccharides, DNA, and proteins

in the biofilm surface

$1 cm$

$1 cm$

Dietrich, L., et al. Journal of Bacteriology (2013)

Structures

Multiple light sources in the instrument
Super-resolution measurements
Non-invasive
No preparation needed

$0.5 mm$

0

2

4

6

8

10

$\Delta z$ ( $\mu m$ )

Central region of a vibrio cholerae biofilm

Surface topography + intensity!

White-light interferometry

inside the colony

Nutrient dynamics

$z$

\frac{\partial c}{\partial t} = D \cdot \frac{\partial^2 c}{\partial z^2} - \lambda \cdot \frac{c}{k+c}

\frac{\partial c}{\partial t} = D \cdot \frac{\partial^2 c}{\partial z^2} - \lambda \cdot \frac{c}{k+c}

c(0, t) = c_0 \quad \quad \quad \dot{c}(h, t) = 0

c(0, t) = c_0 \quad \quad \quad \dot{c}(h, t) = 0

Concentration in the substrate

No flux in the top

Region where cells can grow is finite

\int_0^h \frac{c(z)}{k+c(z)}dz \approx \text{min}(h, L)

\int_0^h \frac{c(z)}{k+c(z)}dz \approx \text{min}(h, L)

Total growth of a colony saturates once they reach a critical length $L$

Diffusion constant

Consumption rate

Monod constant

inside the colony

Nutrient dynamics

$z$

\frac{\partial c}{\partial t} = D \cdot \frac{\partial^2 c}{\partial z^2} - \lambda \cdot \frac{c}{k+c}

\frac{\partial c}{\partial t} = D \cdot \frac{\partial^2 c}{\partial z^2} - \lambda \cdot \frac{c}{k+c}

c(0, t) = c_0 \quad \quad \quad \dot{c}(h, t) = 0

c(0, t) = c_0 \quad \quad \quad \dot{c}(h, t) = 0

Concentration in the substrate

No flux in the top

Region where cells can grow is finite

\int_0^h \frac{c(z)}{k+c(z)}dz \approx \text{min}(h, L)

\int_0^h \frac{c(z)}{k+c(z)}dz \approx \text{min}(h, L)

Total growth of a colony saturates once they reach a critical length $L$

Consumption rate

Diffusion constant

Monod constant

2560

for vertical

biofilm growth

Heuristic model

Empirical data + biophysical insight:

Fits the dynamics across timescales
Captures two growth regimes
$D, \lambda, k \rightarrow L$

\frac{\partial h}{\partial t} = \alpha \cdot \text{min}(h, L) - \beta \cdot h\\

\frac{\partial h}{\partial t} = \alpha \cdot \text{min}(h, L) - \beta \cdot h\\

Diffusion length

Growth rate

Decay rate

Why is the prediction lower?

h_{\text{max}} = \frac{\alpha \cdot L}{\beta}

h_{\text{max}} = \frac{\alpha \cdot L}{\beta}

Growth

Decay

Small overestimation of $\beta$ can lead to underestimating $h_{\text{max}}$

Measuring for 48h is going to dry the plate a bit more, even if we try to minimize it

High-tech containment chamber for timelapse measurements.

Even if the prediction gets corrected with time, residual changes very little!

Do the make sense?

parameters

\frac{\partial c}{\partial t} = D \cdot \frac{\partial^2 c}{\partial z^2} - \lambda \cdot \frac{c}{k+c}

\frac{\partial c}{\partial t} = D \cdot \frac{\partial^2 c}{\partial z^2} - \lambda \cdot \frac{c}{k+c}

$800 \mu m^2 \cdot s^{-1}$

$38 \mu M$

$1.3\cdot10^3 \mu M \cdot s ^{-1}$

E. coli growing in agar -> limited by L-serine

Using literature parameters we obtain

$L = 14.8 \mu m$

And using the interface model

$L = 14.3 \pm 1 \mu m$

2560

Do the make sense?

parameters

\frac{\partial c}{\partial t} = D \cdot \frac{\partial^2 c}{\partial z^2} - \lambda \cdot \frac{c}{k+c}

\frac{\partial c}{\partial t} = D \cdot \frac{\partial^2 c}{\partial z^2} - \lambda \cdot \frac{c}{k+c}

$800 \mu m^2 \cdot s^{-1}$

$38 \mu M$

$1.3\cdot10^3 \mu M \cdot s ^{-1}$

E. coli growing in agar -> limited by L-serine

Using literature parameters we obtain

$L = 14.8 \mu m$

And using the interface model

$L = 14.3 \pm 1 \mu m$

1920

behavior

Long-time

Measure colonies every 2 days
Equilibrium in maximum height
Model height prediction:

$h_{\text{max}} = \frac{\alpha L}{\beta}$
Same behavior, different parameters
Good agreement even early!

What is ?

Profiles are flat. A few cells in amplitude, over thousands of micrometers!

$500 \mu m$

biofilm topography

Using white-light interferometry, we can capture the profiles of a growing colonies for extended periods of time

Width of perpendicular fluctuations in the height profile:

w_l(\mathbf{r}, t) = \langle (h (\mathbf{r}, t) - \langle h(\mathbf{r}, t) \rangle_l )^2 \rangle_l^{1/2}

w_l(\mathbf{r}, t) = \langle (h (\mathbf{r}, t) - \langle h(\mathbf{r}, t) \rangle_l )^2 \rangle_l^{1/2}

$\alpha$

II. Quantifying fluctuations

Self similar systems link the fractal dimension $FD$ and roughness $\alpha$ with:

FD + \alpha = n + 1

FD + \alpha = n + 1

Fractal Dimension FD

Measure of how complexity changes with scale

Surface roughness $\alpha$

are different between strains

Staphylococcus aureus

Bacillus cereus

Eschericia coli

What is leading to the differences between profiles?
How do we characterize the dynamics?
Why does the topography freeze in later alligators?

$2 mm$

$8 \mu m$

Time since inoculation [hours]

$0$

$24$

$48$

Fluctuation dynamics

Are fluctuations ?

$10^4$

$10^3$

$10^2$

$10^1$

scale-free

Wavelengths of $\lambda = L/2$ were removed in each step

$S(k, t) \sim k^{-\nu}$

PSD exhibits $1/f$ spectra
Self organized criticality
Observed in vacuum tubes, human hearts, brain activity, musical concertos.

Testing again

self-affinity

We can test self-affinity using the fractal dimension $D$

$D + H = 2$

$D + H = n +1$ , where $n$ is the base dimension of the system

Topography dynamics as a consequence of growth through a viscoelastic material:

Reach SA while growth is fast
Relax when the colony is tall, and fluctuations get damped

Width of perpendicular fluctuations in the height profile:

With a roughness/Hurst scaling exponent $\alpha, H$ :
And a growth exponent $\beta$ :

w_{\ell}(\mathbf{r}, t) = \langle (h (\mathbf{r}, t) - \langle h(\mathbf{r}, t) \rangle_{\ell} )^2 \rangle_{\ell}^{1/2}

w_{\ell}(\mathbf{r}, t) = \langle (h (\mathbf{r}, t) - \langle h(\mathbf{r}, t) \rangle_{\ell} )^2 \rangle_{\ell}^{1/2}

$w_{\ell}(t) \propto \ell^{H}$

fluctuations

Dervaux, et al. 2014

Martinez-Calvo and Bhattacharjee, et al. 2022

$H$

${\ell}_{\text{sat}}$

$w_{\text{sat}}$

Quantifying

${\ell}$

w_{\text{sat}}(t) \sim \begin{cases} t^{\beta} & \text{for } t < t_{\times}\\ \text{constant} & \text{for } t > t_{\times} \end{cases}

w_{\text{sat}}(t) \sim \begin{cases} t^{\beta} & \text{for } t < t_{\times}\\ \text{constant} & \text{for } t > t_{\times} \end{cases}

Spatial scaling:

Roughness stabilizes at $H \sim 0.8$ .

Biofilm interfaces, have been characterized at $0.6-0.78$

roughening

$w_{\ell}(t) \propto \ell^{H}$

${\ell}$

Paper			Roughness
On growth and form of Bacillus subtilis biofilms	~2	~1.5	0.5-0.7
Morphological instability and roughening of growing 3D bacterial colonies	~1.5	~1	0.67
Yunkerlab-Interferometry	~3	~2.5	0.74-0.84

Temporal scaling:

Saturation width decreases!

Something different

is happening after $t_{\times}$

saturation?

w_{\text{sat}}(t) \sim \begin{cases} t^{\beta} & \text{for } t < t_{\times},\\ \text{constant} & \text{for } t > t_{\times} \end{cases}

w_{\text{sat}}(t) \sim \begin{cases} t^{\beta} & \text{for } t < t_{\times},\\ \text{constant} & \text{for } t > t_{\times} \end{cases}

II

Topography

freezing

Two interface stages:
1. Active
2. Frozen
Qualitative behavior in all interfaces.
No traveling waves
Correlation length $\xi$ increases slowly with time $t$

Interface dynamics and

growth

Fluctuations reach a maximum defined by $w_{\text{sat}}$
Growth saturates when colony reaches critical length $L$
Strong correlation between fluctuation and growth dynamics
Every additional layer, is a
damping element for fluctuations.

characterization of

growth

Pablo Bravo

Topographic

biofilm

QBioS PhD Defense

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