FYS4715 - Biologisk fysikk

Active matter: Lecture notes

What is active matter?

Examples of living active matter

School of fish

E-coli swarming

Cell migration

Single active particles: Self-propelled Brownians

Figure 1: Self-propulsion speed as function of the size of a microswimmer.

Figure 2: Random trajectory of a run and tumble particle (RTP) versus an active Brownian particle (ABP)

Active Brownian

This is like an overdamped (no inertia) Brownian particle which has two kinds of motion:

  1. random meandering like a passive Brownian with a noise amplitude determined by the translational diffusivity, \( D_T \).
  2. self-propulsion with a constant speed, \( v \), but in a random direction \( \theta \). The randomness in the orientation of the self-propulsion has a noise amplitude given by the rotational diffusivity, \( D_R \).

For a motion confined to the x-y plane, the active Brownian is modelled by 3 dynamical variables \( (x(t),y(t),\theta(t)) \) , representing its position and the angle of the self-propulsion speed. The dynamical equations for these variables are:

$$ \begin{eqnarray} \dot x &=& v\cos\theta +\sqrt{2D_T}\xi_x, \quad \langle\xi_x\rangle = 0,\quad \langle\xi_x(t)\xi_x(t')\rangle =\delta(t-t')\\ \dot y &=& v\sin\theta +\sqrt{2D_T}\xi_y, \quad \langle\xi_y\rangle = 0,\quad \langle\xi_y(t)\xi_y(t')\rangle =\delta(t-t')\\ \dot \theta &=& \sqrt{2D_R}\xi_\theta, \quad \langle\xi_\theta\rangle = 0,\quad \langle\xi_\theta(t)\xi_\theta(t')\rangle =\delta(t-t') \end{eqnarray} $$

Overdamped motion

We can rewrite equations of motion in a vectorial form, so that it becomes more clear that this is an overdamped motion, with the drag coefficient \( \gamma \)

$$ \begin{equation} \gamma\mathbf{\dot r} = F\mathbf{\hat e} +\mathbf{\xi} \label{_auto1} \end{equation} $$

Finite-difference equations

These equations are typically solved numerically using discrete time with \( \Delta t \) being the time increment between successive times, \( t_{n+1} = t_n+\Delta t \). The corresponding finite-difference equations are:

$$ \begin{eqnarray} x_{n+1} &=& x_n+v\cos\theta_n\Delta t+\sqrt{2D_T \Delta t}W_{x,n}\\ y_{n+1} &=& y_n+v\sin\theta_n\Delta t+\sqrt{2D_T \Delta t}W_{y,n}\\ \theta_{n+1} &=& \theta_nt+\sqrt{2D_R \Delta t}W_{\theta,n} \end{eqnarray} $$

where \( W_{x}, W_{y}, W_\theta \) are Gaussian-distribution random numbers with zero mean and unit standard deviation which are independently drawn at each time-step.

Run-and-tumble

The run-and tumble model was inspired by chemotaxis of bacteria, i.e. how bacteria move towards or away from their food source.

Figure 3: Run and tumble modes

It alternates between an active state when the particle move with a constant speed v in a given direction ('run') and a passive state when the particle's position is the same by the orientation keeps changing ('tumble'). The change in direction is taken as instantaneous and occurring with a given rate, \( \lambda/\Delta t \) , such that the particle has a linear trajectory between successive tumblings. The "run" state is punctuated at random times by an instantaneous "tumbling" (reorientation). This is modelled as a Poisson point process characterised by the Poisson probability distribution

$$ \begin{equation} P_\lambda(n) = \frac{\lambda^n}{n!}e^{-\lambda}, \label{_auto2} \end{equation} $$

where \( n= 0,1,2,\cdots \) is the number of tumbling events observed in a time interval \( \Delta t \), \( \lambda \) = average number of tumbling events expected in \( \Delta t \)

Notice that \( P(n=0) = e^{-\lambda} \) is the probability of no tumbling in a time interval \( \Delta t \). Thus, we introduce the probability of tumbling in a time interval \( \Delta t \) as: \( P_{tumble} = 1-e^{-\lambda} \) and the probability of running as: \( P_{run} = P(n=0) = e^{-\lambda} \)

Now, we can model the motion of a single bacteria moving in the plane through 4 dynamical variables \( (x,y,\theta,\rho) \), the additional one being the variable \( \rho \) which keeps track of the state of the bacteria (1: run, 0: tumble).

The finite-difference evolution equations for these variables are:

$$ \begin{eqnarray} x_{t+1}&=& x_n+\rho_n v\cos\theta_n \Delta t+\sqrt{2D_T \Delta t}W_{x,n}\\ y_{n+1}&=& y_n+\rho_n v\sin\theta_n \Delta t+\sqrt{2D_T \Delta t}W_{y,n}\\ \theta_{n+1}&=& \theta_n+(1-\rho_n)\Delta \theta_n\\ \rho_{n+1} &=& \begin{cases} 0, \textrm{ with prob. } 1-e^{-\lambda} \\ 1, \textrm{ with prob } e^{-\lambda} \end{cases} \end{eqnarray} $$

where \( W_{x}, W_{y}, W_\theta \) are Gaussian-distribution random numbers with zero mean and unit standard deviation which are independently drawn at each time-step.

Diffusive behavior of self-propelled Brownians

Let's introduce the characteristic timescale of rotational diffusion (meandering) as \( \tau_R = 1/D_R \). On timescales much larger that \( \tau_R \), the self-propelled Brownian will diffuse with an effective diffusivity that is larger than the thermal diffusivity

$$ \begin{equation} D_{eff} = D_T+ \frac{v^2\tau_R}{4} \label{_auto3} \end{equation} $$

Figure 4: Mean square displacement as function of time has three characteristic regimes: i) passive diffusion on very short timescales, ii) intermediate ballistic regime, and iii) diffusion with enhanced diffusivity on long timescales

$$ \begin{eqnarray} \langle(\Delta r)^2\rangle (t)&=& \langle(\Delta x)^2\rangle(t)+\langle(\Delta y)^2\rangle(t)\\ \langle(\Delta r)^2\rangle (t)&=& \begin{cases} 4D_T t+v^2t^2, \qquad t\ll \tau_R\\ (4D_T +v^2\tau_R)t, \qquad t\gg \tau_R \end{cases} \end{eqnarray} $$

Diffusivity coefficient

$$ \begin{equation} D_{eff}= \lim\limits_{t\rightarrow\infty}\frac{\langle|\Delta r|^2\rangle}{4t} = D_T+\frac{v^2 \tau_R}{4} \label{_auto4} \end{equation} $$

Interacting self-propelled Brownians: Vicsek model for swarming

Viscek (Vicsek et al. 1995) published a minimal model for flocking behavior due to alignment interactions between neighboring particles.

For simplicity, we consider active Brownians moving in the plane and each of them is described by their coordinates \( (x^{(i)}, y^{(i)}) \) and orientation of motion \( \theta^{(i)} \), where \( i \) is the particle label.

Figure 5

Finite-difference equations

The finite-difference evolution equations for particle i are:

$$ \begin{eqnarray} x_{n+1}^{(i)} &=& x_n^{(i)}+v\cos\theta^{(i)}_n\Delta t\\ y_{n+1}^{(i)} &=& y_n^{(i)}+v\sin\theta^{(i)}_n\Delta t\\ \theta_{n+1}^{(i)} &=& \theta_n^{(i)}+(\langle\theta_n^{(j)}\rangle_{S_i}-\theta_n^{(i)})+\sqrt{2D_R\Delta t} W_{\theta^{(i)},n} \end{eqnarray} $$

where \( \langle\theta_n^{(j)}\rangle_{S_i} \) is the average orientation of all the active particles in the neighborhood of \( i \)-th particle (see illustration above)

Numerical examples showing the emergence of polar order (flocking):

Viksek model with an interaction radius \( R_1 = 0.1 L \) and higher noise

Viksek model with a larger interaction radius \( R_2 = 0.2 L \) and higher noise

Viksek model with \( R_1 \) and lower noise

Viksek model with \( R_2 \) and higher noise

Interacting active particles in confinement

Active particles with topological interactions and confined to a disk

Emergent nematic order in bacteria colonies

Bacteria are elongated particles with rod-like shapes and when they are closely packed they have a tendency to have a similar orientation by aligning their principal axis long a preferred orientation. In doing so, they break rotational symmetry (i.e. all directions are equally accessible) and form an ordered state with a preferred orientation. This called a nematic order.

Figure 6: Snapshots of a growning bacteria colony in which bacteria self-organise and align their orientation in a preferred direction.

Nematic order commonly found in passive systems, known as liquid crystals. By analogy, active matter nematic order is often called active liquid crystals.

Polar versus nematic order

Figure 7

Polar ordering of epithelial cell layers

Epithelial cells

Wiki: A thin, continuous, protective layer of compactly packed cells with a little intercellular matrix. Epithelial tissues line the outer surfaces of organs and blood vessels throughout the body, as well as the inner surfaces of cavities in many internal organs.

Figure 8

Figure 9

Topological defects

Figure 10: Typical topological defects observed in a liquid crystal sample. With each defect, there is an associated topological charge s, defined as the number of times the vector field winds around a loop encircling the defect once in an anticlockwise sense. Hence (a) and (b) have s = +1, (c) s = −1, (d) s = + 1/2, (e) s=-1/2

Flow and polar ordering

Integer topological defect annihilations drive polar ordering of epithelial monolayers (2023) Emma Lång, Anna Lång, Pernille Blicher, Torbjørn Rognes, Paul Dommersnes, Stig Ove Bøe

Figure 11

a:

Time evolution of polar ordering. The color map represents the angle of the local velocity field relative to the x-axis. The surface of a whole circular monolayer is shown. Scale bar, 1 mm.

b:

Detection and mapping of ±1 topological defects.

h:

Time series showing annihilation of a ±1 topological defect pair. Scale bar, 100 µm.

Active Elastic Solid model

$$ \begin{eqnarray} \dot{{\bf R_i}}&=&V_c{\bf P_i}+\frac{1}{\xi}{\bf F_i} \label{eq:rdot} \\ \dot{{\bf P_i}}&=&\gamma({\bf P_i}\times\dot{{\bf R_i}})\times\dot{{\bf P_i}} \label{eq:pdot} \end{eqnarray} $$

where \( \dot{{\bf R_i}} \) is the cell positions and \( \bf P_i \) the polarity of cell propulsion (direction of propulsion), \( V_c \) is the cell propulsion speed in the absence of elastic force, \( \bf F_i \) the elastic force on the cell, and \( \xi \) the substrate friction. While equation \eqref{eq:rdot} can be considered to express balance of forces, equation \eqref{eq:pdot} gives the polar ordering effect and implies that the direction of \( \gamma \) propulsion is turning towards the direction of the flow with a rate constant .

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