Modelling microbial motion: the case of Chlamydomonas reinhardtii

CIBIO Centre for Integrative Biology

Author: Luca Pizzagalli

Supervisor: Prof. Gianluca Lattanzi

Academic year 2017/2018

Chlamydomonas reinhardtii

Alper Joshua et al. Methods in Enzymology, 2013

Green algae, a single cell eukaryotic organism $\approx 10 \mu m$
Model organism in biology, studied for photosynthesis and ciliary functions
Possible source of useful proteins and bio-fuels

How does it move

Puller micro-swimmer
Breast-stroke motion
Run and tumble:
straight swims alternated with rapid rotations

Alper Joshua et al. Methods in Enzymology, 2013

How to define a model for the motion of C. reinhardtii?

Run and tumble motion
Interaction with obstacles
Results coherent with experiments

The Model

2D Asymmetric dumbbell, a sphere for the cell's body and a sphere for the area covered by the beating flagella

C. reinhardtii with a spherical body and two flagella

Wysocki Adam et al. Phys. Rev. E, May 2015

Equations of motion

$\frac{d\boldsymbol{r}(t)}{dt} = v_0 \boldsymbol{e}(t) + \mu \boldsymbol{F}(\boldsymbol{r},\boldsymbol{e}) + \boldsymbol{\eta}(t)$

$\langle \boldsymbol{\eta}(t) \boldsymbol{\eta}(t') \rangle = 2 k_b T \mu \boldsymbol1 \delta (t-t')$

Forces

$\boldsymbol{F}(\boldsymbol{r},\boldsymbol{e}) = \boldsymbol{F}(\boldsymbol{r}_b) + \boldsymbol{F}(\boldsymbol{r}_f)$

$= -\nabla U_b(\boldsymbol{r}_b) - \nabla U_f(\boldsymbol{r}_f)$

Truncated Lennard-Jones potential:

$U_b(\boldsymbol{r}_b) = 4 \epsilon k_b T \Big[\Big(\frac{a_b}{|\boldsymbol{r}_b - \boldsymbol{r}_w|}\Big)^{12} - \Big(\frac{a_b}{|\boldsymbol{r}_b - \boldsymbol{r}_w|}\Big)^6\Big] + \epsilon$

$U_f(\boldsymbol{r}_f) = 4 \epsilon k_b T \Big[\Big(\frac{a_f}{|\boldsymbol{r}_f - \boldsymbol{r}_w|}\Big)^{12} - \Big(\frac{a_f}{|\boldsymbol{r}_f - \boldsymbol{r}_w|}\Big)^6\Big] + \epsilon$

Direction

$\frac{d\boldsymbol{e}(t)}{dt} = \Big( \frac{\boldsymbol{T}(\boldsymbol{r},\boldsymbol{e}, t)}{\tau} + \boldsymbol{\xi} \Big) \times \boldsymbol{e}(t)$

$\langle \boldsymbol{\xi}_z(t) \boldsymbol{\xi}_z(t') \rangle = \frac{2 k_b T}{\tau_p} \delta (t-t')$

Direction

Alternative expression, more useful for numerical computation

$\frac{d \vartheta(t)}{dt} = \frac{T_z(\boldsymbol{r},\vartheta)}{\tau} + \xi_z(t)$

Direction

$\boldsymbol{T}(\boldsymbol{r},\boldsymbol{e}, t) = \boldsymbol{T}_b + \boldsymbol{T}_f = -c (\boldsymbol{e} \times \boldsymbol{F}_b) + (l-c) (\boldsymbol{e} \times \boldsymbol{F}_f)$

Tumble

$\boldsymbol{T}(\boldsymbol{r},\boldsymbol{e}, t) = \boldsymbol{T}_w(\boldsymbol{r},\boldsymbol{e}) + \boldsymbol{T}_t(t)$

Time between two tumbles: exponential distribution with average $11.2 s$
Tumble duration: Gaussian distribution with average $2 s$
Tumble strength: Gaussian distribution with average $0.75 rad/s$

Parameters

variable
$a_b$
$a_j$
$l$
$c$
$v_0$
$k_b T \mu$
$\frac{\tau_p}{k_b T}$
$\frac{\tau}{k_b T}$
$\epsilon$
$\mu [ t_{\text{s}} ]$
$\mu [\frac{T_t}{\tau} ]$
$\sigma [ \frac{T_t}{\tau} ]$
$\mu [ t_{\text{t}} ]$
$\sigma [ t_{\text{t}} ]$

value	m. u.
$5.0$	$\mu m$
$7.5$	$\mu m$
$7.5$	$\mu m$
$0.0$	$\mu m$
$60 - 110$	$\mu m/s$
$??$	$\mu m^2/s$
$??$	$s$
$0.15$	$s$
$10$	$1$
$11.2$	$s$
$0.75$	$rad / s$
$0.75$	$rad / s$
$2.0$	$s$
$1.5$	$s$

Experimental Validation

Vasily Kantsler et al. PNAS, January 2013

Mean square displacement from experiment in open space

Short time: ballistic behavior
Long time: diffusion behavior

Experimental Validation

Tanya Ostapenko et al. Phys. Rev. Lett., February 2018

Radial probability $P(r)$ in confined environment

$P(r) = \frac{\frac{h(r)}{2 \pi r \Delta r}}{\int_{0}^{R}\frac{h(r')}{2 \pi r' \Delta r} d r'}$

Numerical Simulation

Experiment

Tanya Ostapenko et al. Phys. Rev. Lett., February 2018

Simulation

github.com/LucaPizzagalli/swimmers-brownian-simulation

Experimental Validation

Experiment

Simulation

Experimental Validation

Experiment

Simulation

Parameters

variable
$a_b$
$a_j$
$l$
$c$
$v_0$
$k_b T \mu$
$\frac{\tau_p}{k_b T}$
$\frac{\tau}{k_b T}$
$\epsilon$
$\mu [ t_{\text{s}} ]$
$\mu [\frac{T_t}{\tau} ]$
$\sigma [ \frac{T_t}{\tau} ]$
$\mu [ t_{\text{t}} ]$
$\sigma [ t_{\text{t}} ]$

value	m. u.
$5.0$	$\mu m$
$7.5$	$\mu m$
$7.5$	$\mu m$
$0.0$	$\mu m$
$10$	$1$
$70$	$\mu m/s$
$14$	$\mu m^2/s$
$2.0$	$s$
$2.0$	$s$
$11.2$	$s$
$0.75$	$rad / s$
$0.75$	$rad / s$
$2.0$	$s$
$1.5$	$s$

Dependence on parameters

Propulsion Speed

Msd in open space

$P(r)$ in confinement

Radial probability in confined space as a function of speed.

Rotational Axis

Msd in open space

$P(r)$ in confinement

Rotational Axis

Tumble Strength

Msd in open space

$P(r)$ in confinement

Radial probability in confined space as a function of tumble strength.

Let us consider now the influence of the tumble on the statistical properties of the micro-swimmer's movement. Different simulations have been performed for different values of T_t, the torque representing the tumble. From the figure we can see that the tumble starts having an influence on the mean-square-displacement only after about one second from the start; this is caused by the fact that in the simulation the microswimmer starts in a non-tumbling configuration, and only after a random time the first tumble begins. A stronger torque means a more drastic reorientation of the cell and this is reflected in a minor mean-square displacement. The tumble has also an effect on the radial probability. The persistence time near the wall decreases with the increasing strength of the tumble as a stronger tumble means higher probability of reorienting the cell away from the wall. Analogous effects on both mean-square-displacement and radial probability can be expected to induce a variation in the tumble duration or in the frequency of tumbling events.

Translational Noise

Msd in open space

$P(r)$ in confinement

Radial probability in confined space as a function of translational noise strength.

Rotational Noise

Msd in open space

$P(r)$ in confinement

Radial probability in confined space as a function of rotational noise strength.

Collective Behavior

Four components for the force:

$\boldsymbol{F}(\boldsymbol{r}_1,\boldsymbol{e}_1,\boldsymbol{r}_1,\boldsymbol{e}_2) = \\ -\nabla U_{bb}(\boldsymbol{r}_{1, b},\boldsymbol{r}_{2, b}) - \nabla U_{ff}(\boldsymbol{r}_{1, f},\boldsymbol{r}_{2, f}) \\ - \nabla U_{bf}(\boldsymbol{r}_{1, b},\boldsymbol{r}_{2, f}) - \nabla U_{bf}(\boldsymbol{r}_{1, f},\boldsymbol{r}_{2, b})$

from truncated Lennard-Jones potentials

Diffusion

Linear density of a diffusion of C. reinhardtii in a test tube, 1 minute after centrifugation.

Experiment

Simulation

Experimental measurements of Chlamydomonas' macroscopic diffusion are available from the work polin2009chlamydomonas. A suspension of 1.5 ml of Chlamydomonas is transferred in a test tube of size 20 x 12.2 x 4 mm and centrifuged at 350 g for 2.5 min, causing the sedimentation of the suspended cells. The figure reports the measurements of the cells' linear density distribution across the test tube after one minute form the end of the centrifugation process. I simulated one minute of 10_000 Chlamydomonas' cells diffusing from a concentrated initial state, with a time-step of 10^-3 s. The picture reports the linear concentration at different times of the micro-swimmers' suspension according to the simulation. The final distribution is qualitatively similar to the experimental measurement. However, the max value of the simulated distribution is not at the absolute bottom of the test tube only because of the starting condition. Instead, for the experiment, the reason explaining why the max density is not exactly located at the bottom is to be found in the physical traits of C. reinhardtii.

Conclusions

The model developed is powerful enough to reproduce:

chlamydomonas reinhardtii moves beating two forward flagella

Ballistic and diffusive behavior in open space
Long retention times when near convex walls
Diffusion of cells in a suspension

In this work I developed a mathematical model of C. reinhardtii. The model formalizes the eukaryotes' motion taking into account the propulsion coming from the flagella activity, the perturbation due to the thermal noise, the tumbling activity, and the interaction between the cell and a surface. The comparison between the numerical simulations and empirical data shows that the developed model is able to qualitatively reproduce some fundamental properties of the motion of the cell and its interaction with obstacles: - Looking at the mean-square-displacement as function of time, the model is able to reproduce both the ballistic behaviour typical of the first instants of motion and the subsequent transition to a diffusive motion when the direction the cell is facing has been randomized by the rotational noise and the tumble events. - The compatibility as been observed also with experiments of a single cell in a confined environment. The plot of the radial probability resulting from the numerical simulations well captures the facts that most of the time is spent by the cell near the surface. - Finally, the plot of the time evolution of the simulation of a multitude of active particles in the same environment shows that it is also possible to capture some the collective quality of the Chlamydomonas' motion. I then proceeded testing the robustness of the model by changing the value of its parameters. A significant dependence of the result on all the parameters is observed. For this reason, and having in mind the variability and uncertainty of some parameters of the model, we can expect that quantitative predictions based on this model can show some discrepancies with empirical data. The last comparison between the diffusion of thousands simulated active-particles and the diffusion of the real cells of C. reinhardtii in a suspensions shows a first limitation in the description power of the model. The real Chlamydomonas exhibits gravitaxis while for the mathematical model here developed the space is isotropic. This introduces the opportunity for future developments and improvements. The model here developed does not exactly describe all the aspects of the behavior of C. reinhardtii. However an extension of such model could capture the way C. reinhardtii perceives the environment and adapts to the different conditions, particularly reproducing gravitaxis, chemotaxis and phototaxis.

Conclusions

The quantitative results are in good agreement with the experimental values, but:

Some parameters are tuned to experiments
Results show a significant dependence on the parameters used

Conclusions

Further extensions could take into account how the cell perceives the environment and modifies its behavior accordingly, including:

Chemotaxis
Gravitaxis
Phototaxis

Modelling microbial motion: the case of Chlamydomonas reinhardtii

Chlamydomonas reinhardtii

How does it move

How to define a model for the motion of C. reinhardtii?

The Model

Equations of motion

Forces

Direction

Direction

Direction

Tumble

Parameters

Experimental Validation

Experimental Validation

Numerical Simulation

Experimental Validation

Experimental Validation

Parameters

Dependence on parameters

Propulsion Speed

Rotational Axis

Rotational Axis

Tumble Strength

Translational Noise

Rotational Noise

Collective Behavior

Diffusion

Conclusions

Conclusions

Conclusions

Thank you for your attention