Scaling in the Physarum Amobae
during locomotion:
size vs shape, speed, rhythm
Shigeru Kuroda
Research Institute for Electrical Science,
Hokkaido University
MIMS Workshop “反応拡散現象にみられる境界層とその周辺の数理”,
Nov 28, 2014

Outline•Backgournd•Self-organized Phenomena as Directional locomotion ofAmoeba cell
•Various size in Physarum amoebae•Experimental•Methods•Results•A Math. Model•Amoeba locomotion with large fluctuated speed as
Traveling wave-like solution in non-autonomous R-Dsystems

Outline•Backgournd•Self-organized Phenomena as Directional locomotion ofAmoeba cell
•Various size in Physarum amoebae•Experimental•Methods•Results•A Math. Model•Amoeba locomotion with large fluctuated speed as
Traveling wave-like solution in non-autonomous R-Dsystems

Biological scaling (allometry)•The study of the relationship of body size to the othercharacteristics (ex) basal metabolic rate, cardiac rate, size ofbody parts, speed, shape, ...).

•referred as allometry in zoology.

Why are the scalable properties
important?
It does not involve chance but instead tells us about thephysical laws behind biological behaviour.

In particular,
the observation of scaling laws that hold across size
differences of several orders of magnitude may lead to
substantial revisions in the understanding of not only thefield of interest but also in many other fields of study.

metabolic rate

3/4
mm

Slime mould(Physarum Policepharum)

Background•Multinucleus single-cell organism with amoeboid
movement•Nuclear division without cell division occur per
10 hour••Experimental convenience of size manupulation:-5
Large variation of cell size: 10 m ~ 1m

fuse and cut

Growth of single-nucelus unicelluar organism

Nuclear division with cell division

Growth of multinucleus unicelluar organism(Physarum)

Nuclear division without cell division

Experimental convenience of size manupulation:

fuse

cut! Body size can thus be regarded as a system parameter !

Background

Directional movement of the cellas self-organized phenomenon

Plasmodium ofTrue slime mouldNote. In our body, there are many kinds of cells using amoeboid motion.
(ex) White blood cell (leukocyte), Epithelial cell, Cancer

speed x 3600
10mm

Small piece taken from a large Physarum amoeba

Background

Slime mould with various size
during locomotion

100μm

500mm

1mm

1cm

Scaling study of cell movement
using Slime mouldbody length• Static allometry
the size dependency of the time-averaged characteristics•Ontogenetic allometry
the characteristics changes throughout its development

Outline•Backgournd•Self-organized Phenomena as Directional locomotion ofAmoeba cell
•Various size in Physarum amoebae•Experimental•Methods•Results•A Math. Model•Amoeba locomotion with large fluctuated speed as
Traveling wave-like solution in non-autonomous R-Dsystems

Experimental environment
１w%agar, room temp. 24±1!, the humidity
inside the apparatus 90%RH
more than six hour measurement of one
directional locomotion
Smaller slime moulds (< 0.1μL (n=13))
allowed to move freely on agar plate (φ=9cm)
Larger slime moulds ( > 0.1μL (n=37))
placed on narrow agar lanes (width 0.5, 1, 2, 4,or 8 mm) in order to limit the num of advancing
front to one or to confine locomotion to 1-dim.

Systematic preparation of cells withdifferent sizes from a giant cell
For medium and larger size (> 0.1!L)250(mm)cutlw

PET film

cutagar (1w%)
width of each lane lw = 1, 2, 4 or 8 [mm]cutwet cellophane

For smaller size(< 0.1!L)Measurement of thickness
Optical density (x,t)!"#$%

Caribulation

thicknessh(x,y,t)Smaller slime moulds (< 0.1μL (n=13))
allowed to move freely on agar plate (φ=9cm)

Coordinate system and longitudinal shapehmax : the max thickness of frontal partxL : body length
xmax : length of very frontal tip
dV :hʹ′(x, t)dVLongitudinal coordinate andits perpendicular axis.

Characterisation of longitudinal shape

dinate and longitudinal shape. The plots are not from real data. (a) introducti
perpendicular axis (b) characterization of longitudinal shape of plasmodium:he effective body length (xL ), and the maximum thickness of frontal part (hmax ).
ʹ′(x, t) = maxy∈l(x,t) h(y, t)hont edge. The abscissa axis is the longitudinal axis of the moving longitude coo

Larger slime moulds ( > 0.1μL (n=37))
placed on narrow agar lanes (width 0.5, 1, 2, 4, or 8mm) in order to limit the num of advancing front toone or to confine locomotion to 1-dim.

Regular oscillations in thickness with aperiod of 60- 120s that involves
protoplasmic flow inside the body

Speedx100

Speedx100

Averaging of variables over intrinsic period (T0) ofthickness oscillation

Averaging over T0

Outline•Background•Self-organized Phenomena as Directional locomotion ofAmoeba cell
•Various size in Physarum amoebae•Experimental•Methods•Results•A Math. Model•Amoeba locomotion with large fluctuated speed as
Traveling wave-like solution in non-autonomous R-D
systems•Static allometry
the size dependency of the time-averaged characteristics

•Background•Self-organized Phenomena as Directional locomotion ofAmoeba cell
•Various size in Physarum amoebae•Experimental•Methods•Results•A Math. Model•Amoeba locomotion with large fluctuated speed as
Traveling wave-like solution in non-autonomous R-D
systems

velocity at small d can be caused by the damage due to the amputation.
This result Localizationsuggests that the rhythmiccontraction
isnotamotiveforce
ofof the locomotion
locomotion and extension of the frontal tip, and the frontal locomotion willengineinatthefrontal partbe driven by a process
localized
the tip.

b)a)

10.8

va /v0space (mm)

60.6

4

0.4

2
0.2

00010time (min)

200

12

34

56d (mm)(from
Takagi,S.theet allocomotion
2007)
Fig. 3. Amputation of the tip
to examine
activity of the frontal tip. (a)Space-time plot of thickness profile of a plasmodium directionally migrating in a narrow lane. The thickness is represented by grayscale; darker color corresponds to thicker

Traveling wave in reaction diffusion systemwith slow modulationbody formation
withexponential relaxation
chemical oscillation
withslow modulation (γ)

Discussion
Scalable vs. Non-scalable properties
What is the origin of the difference between them? One probable reason is that:the size-independent properties result from biochemical processes
because chemical reaction speed mainly depends not on the bodysize itself but on the chemical characteristics of the substances
involved.
the size-dependent properties may result from physico-mechanicalprocesses. If so, they are likely to be quite general properties that
hold for a wide range of organisms.