Download Fitting fibrils: Modelling cell wall development in plants

Fitting Fibrils
A geometrical approach to plant cell wall development
• Introduction to plant cell wall morphology
• The current paradigm
• The geometrical theory
Bela Mulder
FOM Institute AMOLF
Amsterdam, NL
• Results
• Conclusions
Anne-Mie Emons
Miriam Akkerman
Plant Cell Biology
Wageningen University, NL
Why study the cell wall ?
Cell walls provide protection
and allow plants to exploit
turgor pressure to raise
themselves against gravity
• Plants make up 99% of the biomass of
earth.
• 10% of this biomass is fixed in plant cells.
• Important source of raw materials: wood,
paper, fibers …
Zooming in on the cell wall
Example cell: root hair
Cross-section
Surface section
Shadow-cast EM image
Cellulose microfibrils
in a polysaccharide
matrix
2-4 nm
CMF
Cell wall textures are regular
The helicoidal cell wall
Real world analogues
Fibre-laminates are both
tough and flexible
Helicoidally
wound strings
pack efficiently
CMF synthases or “rosettes”
The mechanism of CMF synthesis
Certain:
CMF synthases channel
UDP glucose from inside
the cell, and “spin” the
CMF.
Brown et al. (1997)
Arioli et al. (1999)
Plausible:
The CMF synthases are propelled
forward by the polymerization
force and move in the plasma
membrane.
Motion from chemical energy: Polymerisation ratchets
The current paradigm
The so-called microtubule/microfibril paradigm (Giddins&Staehelin [1991])
The CMF synthases are “guided” by the cortical microtubules.
However …
• The hypothesis is mainly supported by the the co-alignment of
CMFs and MTs in expanding cells, where forces are exerted.
• In many non-expanding cells there is no co-alignment between
MTs and CMFs
(Emons [1983,->])
• New Arabidopsis mutants show normal wall development even
when the cortical MT organization is disrupted
(Wasteneys et al.)
• It begs the question of how the cortical MTs are (re)organized.
Background to the geometrical model
CMFs are deposited by CMF
synthases that move in the
plasma membrane.
CMF
existing wall
Deposition takes place in the
limited space between the cell
membrane and the already
extant wall.
cell interior
synthase
membrane
• The CMFs appear closely
packed with a spacing of ~20nm
• CMFs are long L >> 1 m
Ingredient 1: Geometry
track of synthase
microfibril
number of
microfibrils
Geometrical “close packing” rule
(Emons, 1994)
Nd
sin  
2R
cylindrical cell
membrane
Production of synthases
cell wall
membrane
vesicle
exocytosis
synthase
Golgi apparatus
Ingredient 2: space
New synthases created in localised insertion domains
along the cell by the Golgi-apparatus and brought to the
plasma membrane by exocytosis of Golgi-vesicles
L
 (N , t)
rate of synthase creation
depends on number of
synthases already present
=N
constraint
 (N , t)  0
N  N max 
2R
d
Ingredient 3: time
insertion domain moves with velocity v
possible sources of movement:
v
•Cytoplasmic streaming: physical
transport of Golgi apparatus
•Calcium waves: activation/deactivation
of exocytosis
synthase is “born” ( t = 0 )
w
synthase moves
with linear speed w
synthase “dies” ( t = t†)
Putting it all together: a developmental model
Fundamental variable:
N(z,t) = the density of active
synthases at given
location along the cell
geometrical rule
z
dz
sin  ( z, t ) 
N ( z, t )d
2R
Desired result:
(z,t) = the local angle of
deposition of microfibrils
i.e. the cell wall texture
Dynamics of the local synthase density
sources of change
motion of synthases
birth and death of synthases
The evolution equation for the synthase density
N ( z, t ) wd
N ( z, t )

N ( z, t )
  ( N , z, t )   † ( N , z, t )
t
2 R
z
motion of the synthases
activation
deactivation
•The formula make all our assumptions operational.
•It can be used as a “virtual laboratory” in which
“experiments” are performed under different conditions =
values of the parameters of the model.
Results I: the helicoidal wall
Depends on matching of
the size and the speed of
the insertion domain and
the synthase production
rate to the synthase life
time.
Results II: the crossed polylammelate wall
Essentially a helicoidal wall in the case
that the synthase production is initially
very fast, leading to an alternation
between layers with a low and a high
microfibril angle
Results III: the helical wall
Results when the lifetime of synthases is much larger than the
time it takes the insertion domain to travel a distance equal to its
length. Most common wall type of wood.
Results IV: the axial wall
In essence a helical wall with a large
microfibril angle. Highly likely when the
radius of the lumen of the cell is small and
hence the maximum number of CMFs that
can be accommodated is small.
… But is it true ?
Experimental verification:
• Identification of insertion domains.
(Miriam Akkerman, Wageningen)
• Direct visualization of cellulose synthesis and synthase dynamics
in vitro
(FOM/ALW Physical Biology programme II, vacancy)
• GFP tagging of synthases
(exploring collaboration with PRI)
Theoretical elaboration:
• Study the role of physical interactions between synthases
(FOM/ALW Physical Biology programme II, vacancy)
• Generalization to cells with inequivalent facets, cell wall
deposition at the poles of cells, … (future)
Do insertion domains exist?
Dynamics of GFP-tagged Golgi
What is the physical origin of the CMF packing?
Interactions between the
CMFs-synthases:
•Hydrodynamical:
? unknown
•Fluctuation induced (Casimir):
 attractive
•Elastic:
repulsive
Conclusions
• The geometrical theory provides a unified conceptual
framework for understanding cell wall architecture
• It can describe the formation of all known cell wall types
• It is a quantitative model that explicitly allows experimental
verification/falsification.
• Is an example of fruitful interaction between biology and
theoretical/computational sciences.
The evolution equation for the rosette density
N ( z, t )
N ( z, t )
 W ( z, t )
  ( N , z, t )   † ( N , z, t )
t
z
motion of the rosettes
activation
deactivation
wd
N ( z, t )
Local axial speed: W ( z , t )   w sin  ( z , t )  
2 R
Solutions of the model
Helicoidal case, single Insertion Domain, =0
Conditions for helicoid:
 † 1
    12
Solution:



arcsin  2  2   


 ( )  
  arcsin  2  4   2  2   




1 

0    1 
2





1 
1
1 
     2  

 2 

Full solution: “gluing” together a train of Insertion
Domains
v
inter domainspacing
v
v
1
  2 
2