Summary: Active materials can self-organize in many more ways
than their equilibrium counterparts. For example, self-propelled
particles whose velocity decreases with their density can display
motility-induced phase separation (MIPS), a phenomenon building on a
positive feedback loop in which patterns emerge in locations where
the particles slow down. Here, we investigate the effects of
intrinsic fluctuations in the system's dynamics on MIPS. We show
that these fluctuations can lead to transitions between metastable
patterns. The pathway and rate of these transitions is analyzed
within the realm of large deviation theory, and they are shown to
proceed in a very different way than one would predict from
arguments based on detailed-balance and microscopic
reversibility.
Introduction
Bacteria show complex collective behaviour. For example, bacteria such
as E. Coli
- are capable of active propulsion, i.e. have a free-swimming
(planktonic) stage.
- hey are capable of sensing their environment through quorum
sensing, density-dependent gene regulation.
- They stick to surface to form biofilms, high density colonies.
- They exhibit cyclic/time-periodic behaviour: Biofilm formation,
maturation, dispersion, planktonic stage.
The goal of this project is as follows:
Main Problem: Is it possible
to describe similarly complex life-cycles as
emergent behaviour of a large number of simple
individual agents subject to a small number of
collective rules?
Continuum description of motile bacteria
We model the motion of motile reproducing microorganisms with simple
behavioural rules. The self-propulsion is modelled as active
Brownian motion. For position $X\in\Omega\subset\mathbb{R}^d$, direction $\hat n \in S^{d-1}$, and location dependent swim speed $v(X)\in\mathbb{R}$
$$\dot X = v(X) \hat{n}\,,$$
$$d\hat n = \tau^{-1/2} P\circ dW\,,\qquad P=\textrm{Id} - \hat n \hat n^T$$
The direction vector diffuses on $S^{d-1}$ with tumbling rate
$\tau^{-1}$. The fast tumlbing limit $\tau\to0$ yields Brownian motion
$$ dX = \sqrt{2 D(X)} \circ dW $$
with density dependend diffusivity $D(x) = v^2(x)$. Now consider $N$
such particles with position $X_i, i\in{1,\dots,N}$. To model quorum sensing, introduce scale $\delta$ over
which particles feel each other's influence,
$$dX_i = \sqrt{2 D(\rho_{N,\delta}(t,X_i))}\circ dW_i$$
with
$$\rho_{N,\delta}(t,x) = \int_\Omega \phi_\delta(x-y)\rho_N(t,y)\,dy,\qquad \rho_N(t,x) = \frac1N \sum_{j=1}^N \delta(x-X_j(t))$$
In the limit $N\to\infty$, through a Deans-type argument, this yields a closed integro-differential equation for $\rho_N\to\rho$,
$$\partial_t \rho = \nabla\cdot(D(\rho_\delta) \nabla\rho +\tfrac12 D'(\rho_\delta)\nabla \rho_\delta),\qquad \rho_\delta(t,x) = \int_\Omega \phi_\delta(x-y)\rho(t,x) \,dy$$
in the sense that
$$\forall\ \varepsilon,T>0:\quad \lim_{N\to\infty} \mathcal P\Big(\sup_{0\le t\le T}\Big| \frac1N \sum_{j=1}^N f(X_j(t))-\int_\Omega \rho(t,x) f(x)\,dx \Big| >\varepsilon \Big)=0$$
To make this model closed in $\rho(t,x)$, consider $D(\rho)=D_0
e^{-\rho}$, and expand in $\delta\ll1$,
$$\rho_\delta(x) \approx \rho(x) + \tfrac12 \delta^2 \partial_x^2 \rho(x)\,.$$
Then we obtain an effective diffusion equation
$$\partial_t \rho = \nabla \cdot (D_e(\rho)\nabla\rho - \delta^2 \rho D(\rho) \nabla\Delta \rho)$$
with diffusivity
$$
D_e(\rho) = D(\rho) + \tfrac12 D'(\rho)\rho\,.
$$
Despite the microscopic model being not reversible, the continuum
model restores detailed balance
$$
\partial_t\rho = \nabla\cdot(\rho D(\rho) \nabla(\delta E/\delta \rho))
$$
with free energy
$$
E(\rho) = \int_\Omega (\rho\log\rho - \rho + f(\rho) + \tfrac12 \delta^2 |\nabla \rho|^2)\,dx\,,\quad f'(\rho)=\tfrac12 \log D(\rho)
$$
In other words, the model is a $(\rho D(\rho))$-Wasserstein gradient
flow. This model demonstrates the effect of motility induced phase
separation: While the microscopic diffusivity $D(\rho)$ is strictly
positive, this is no longer true for the effective diffusivity
$D_e(\rho)$! For example, for $D(\rho) = D_0 e^{-\rho}$, the effective diffusivty becomes
$$D_e(\rho) = D_0 (1-\tfrac12\rho) e^{-\rho}$$
As a consequence, homogeneous configurations $\rho(x) = \bar\rho$ are
only stable, if $\bar\rho<2$. If instead $\bar\rho>2$ phase
separation occurs, as depicted in the animation. This is the
mentioned feedback loop: Accumulation induced slowdown and
slowdown-induced accumulation.
We now additionally introduce a second behavioral rule for the active
agents: Reproduction and competition. This is models as a
Poisson process
$$A\stackrel{\lambda_r}{\longrightarrow} A+A\qquad\lambda_r=\alpha\qquad\text{(reproduction)}$$
$$A+A\stackrel{\lambda_c}{\longrightarrow} A\qquad\lambda_c=\alpha/\rho_0\qquad\text{(competition)}
$$
with carrying capacity $\rho_0$ and timescale $\alpha$. In the limit $N\to\infty$, this effectively leads to logistic growth
$$\partial_t \rho = \nabla \cdot (D_e(\rho)\nabla\rho - \delta^2 \rho D(\rho) \nabla\Delta \rho) {\color{red} + \alpha \rho (1-\rho/\rho_0)}
$$
Interestingly, now, phase separation will eventually occur if
$D_e(\rho_0)<0$, even if currently $D_e(\bar\rho)>0$. In other words,
the system will drive itself into instability. In particular, we
identify three regimes, with spinodal density $\rho_S$ and critical
density $\rho_c$:
- if $\rho_0 < \rho_S$, only the homogeneous solution will be stable,
and no instability occurs (homogeneous phase)
- if $\rho_S < \rho_0 < \rho_c$, instead limit cycles occur, where
reproduction induces motility induced phase separation, but the
resulting dense bacterial clusters are unsustainable under the
current carrying capacity, and thus slowly die out.
- if $\rho_c < \rho_0$, bacterial clusters become (meta-)stable.
This behavior is reminiscient of the biofilm-planktonic lifecycle:
Bacteria occur in a diluted planctonic phase, until a critical
density is reached. They then stick to walls to form a biofilm,
which over time slowly dissolves back into the planktonic
phase. The corresponding phase-diagram in carrying capacity $\rho_0$
and inverse quorum sensing range (equivalently, domain-size),
$\delta^{-1}$, is shown in the figure.
Relevant publications
- T. Grafke, M. Cates, and E. Vanden-Eijnden, "Spatiotemporal
Self-Organization of Fluctuating Bacterial
Colonies",
Phys. Rev. Lett. 119 (2017), 188003
(link)
