A Process Model of Actin Polymerisation

1
A Process Model of Actin Polymerisation
Luca Cardelli 1, 4 Emmanuelle Caron 2,
4 Philippa Gardner 3, 4 Ozan
Kahramanogullari 3, 4 Andrew Phillips 1
1. Microsoft Research Cambridge 2. Centre for
Molecular Biology and Infection, Imperial College
3. Department of Computing, Imperial College 4.
Centre for Integrative Systems Biology at
Imperial College 12.07.2008 From Biology to
Concurrency and back, Reykjavik
2
The Role of Actin in Cells
Actin is one of the most abundant proteins in
cells. Actin forms one of the three major
cytoskeletal networks in eukaryotic cells.
Actin dynamics plays a key role in cellular
activities such as cell motility and
phagocytosis, e.g., actin polymerisation close to
the cell membrane results in protrusive activity
in phagocytosis.
Actin polymerisation is highly regulated with a
variety of actin binding proteins and in response
to the activation of intracellular signalling
pathways.
3
Process Models of Actin and Geometric Plotting
Differential equation models of actin treat every
filament size as a distinct species. In these
models, because of the representational
boundaries, further aspects of actin dynamics are
often ignored or over-simplified.
It remains a challenge to develop accurate
compositional models of actin.

• We present compositional process algebra models
  of actin polymerisation.
• These models can be easily extended with aspects
  of actin dynamics.
• We introduce a geometric representation of
  process models,
• and apply the developed techniques and tools to
  our actin model.

4
Biology with Processes in SPiM Complexation
new share_at_1.0chan(chan) let A() ( new e_at_1.0
chan() !share(e) Ab(e) ) and Ab(echan)
!e A() let B() ?share(e) Bb(e) and
Bb(echan) ?e B()
A
B
A
B
B
A
B
share(e)
e
Ab
Bb
Ab
Bb
e
5
A Simple Polymer Model
We model complexation of single monomers each
state of a monomer is given by a process.
Af
Af
new c_at_0.116chan(chan) let Af() (new
rht_at_0.0027chan do ?c(lft) Al(lft) or
!c(rht) Ar(rht) ) and Al(lftchan) (new
rht_at_0.0027chan do !lft Af() or
!c(rht) Ab(lft,rht) ) and Ar(rhtchan) ?rht
Af() and Ab(lftchan, rhtchan) !lft Ar(rht)
3
1
?rht
!lft
!c(rht)
?rht
?c(lft)
?c(lft)
4
Al
Ar
Al
Ar
2
!lft
!c(rht)
Ab
Ab

• Association of two monomers.
• Association of monomers at the barbed end (left).
• Dissociation of monomers from the pointed end
  (right).
• Dissociation of dimers.

6
A Simple Polymer Model
Af
Af
Af
Af
Af
Af
Ar
Al
Ar
Al
Al
Ar
Al
Al
Ar
Al
Al
Ab
Ab
Ab
Ab
Ab
Ab
7
A Branching Polymer Model
The branching formation of actin filaments is
initiated by the Arp2/3 complex. Arp2/3 complex
anchors the pointed end of the future daughter
filament to the mother filament. We model this
as a monomer which has three binding sites.
Af
Af
Rf
Al
Ar
Al
Ar
Am
Am
Rr
Rl
Arm
Ab
Alm
Arm
Ab
Alm
Rb
Abm
Abm
8
A Simpler Branching Polymer Model
Adding a new binding site to a monomer increases
the number of states from 4 to 8. Not all of the
newly-added 4 states contribute extensively to
the simulation dynamics. We remove some of the
states in order to obtain a simpler model.
Af
Af
Af
Rf
Al
Ar
Al
Ar
Al
Ar
Am
Rr
Rl
Ab
Ab
Ab
Alm
Arm
Rb
Arb
Arb
Arb
9
Comparing the two Models
We compare these two models by using quantitative
data (Alberts Odel, 2004).
8 states
5 states
These two models behave similarly.
10
Capping Proteins
Capping of the barbed end stops the growth of a
filament. This provides a control mechanism in
actin dependent events.
Af
Ar
Al
Cf
Ab
AlC
Cb
Arb
Cameron et al. 2000
11
Capping Proteins and Branching Actin

• We combine in a single model
• branching filament,
• capping,
• (de)polymerisation at both barbed and pointed
  ends.

Af
Af
Rf
Ar
Al
Ar
Al
Rl
Rr
Ab
Ab
AlC
AlC
Cf
Rb
Arb
Arb
Cb
12
Comparison with Simple Branching Actin
We compare these two models by using quantitative
data (Alberts Odel, 2004). We start the
simulation with 1200 Actin monomers, 30 Arp2/3
and 100 capping proteins.
branching
combined model
Capping proteins reduce the number of left-bound
monomers.
13
An Actin model with Multiple Layers
We extend the model with the role played by
ADP/ATP.
ATP-actin hydrolyses to ADP-Pi-actin, which can
then evolve to ADP-actin. Exchange of ADP with
ATP is fast in the cytosol.
AfT
AfD
AfPi
AlT
AlPi
ArD
ArPi
ArT
AlD
ATC
APC
ADC
Rf
Rr
AbPi
AbD
AbT
Rl
Rb
ARD
ARPi
ART
14
Comparison with Single Layer Model
Single layer model
Multiple layer model
15
Geometric Plotting SPiM Extension

• The simulation results output the change in the
  number of the processes.
• However, it is often beneficial to have geometric
  representations,
• e.g., the change in actin filament meshwork in
  geometric space.
• We encode the geometric information, i.e.,
  coordinates, direction vectors, as process
  parameters.
• Example ... Ab(xfloat, yfloat,x1float,
  y1float, lftchan, rhtchan)

• We apply linear algebra operations to alter
• the geometry, e.g., rotation matrix for 70
  degrees rotation of the daughter actin filament
  at the branching.

Weeds Yeoh, 2001
16
Geometric Plotting SPiM Extension

• We extend the SPiM tool to output event traces,
  and to filter and plot these events. For every
  event, we filter

• its time,
• the species that it consumes,
• the species that it produces.

0.214211814049 Af() Al(59.188584,47.660896,0.76214
6,0.623574,rht5391) --gt Arb(59.188,47.660,0.762
,0.623,rht5391,rht5653) Al(59.950,48.284,0.762,0
.623,rht5653) 0.214216818695 Rf()
Arb(32.3,58.3,0.77,0.63,rht5183,rht5215) --gt
Arbb(32.3,58.3,0.77,0.63,e5655,rht5183,rht5215)
Rl(32.3,58.3,0.9702,0.196,e5655) 0.214229554433
Af() Al(40.9556,18.988,0.9702,0.196,rht5029)
--gt Arb(40.955,18.98,0.97,0.196,rht502,rht5657)
Al(39.985,18.792,0.97,0.196,rht5657)
0.214211814049 Af() Al(59.188584,47.660896,0.76214
6,0.623574,rht5391) --gt Arb(59.188,47.660,0.762
,0.623,rht5391,rht5653) Al(59.950,48.284,0.762,0
.623,rht5653) 0.214216818695 Rf()
Arb(32.3,58.3,0.77,0.63,rht5183,rht5215) --gt
Arbb(32.3,58.3,0.77,0.63,e5655,rht5183,rht5215)
Rl(32.3,58.3,0.9702,0.196,e5655) 0.214229554433
Af() Al(40.9556,18.988,0.9702,0.196,rht5029)
--gt Arb(40.955,18.98,0.97,0.196,rht502,rht5657)
Al(39.985,18.792,0.97,0.196,rht5657)
0.214211814049 Af() Al(59.188584,47.660896,0.76214
6,0.623574,rht5391) --gt Arb(59.188,47.660,0.762
,0.623,rht5391,rht5653) Al(59.950,48.284,0.762,0
.623,rht5653) 0.214216818695 Rf()
Arb(32.3,58.3,0.77,0.63,rht5183,rht5215) --gt
Arbb(32.3,58.3,0.77,0.63,e5655,rht5183,rht5215)
Rl(32.3,58.3,0.9702,0.196,e5655) 0.214229554433
Af() Al(40.9556,18.988,0.9702,0.196,rht5029)
--gt Arb(40.955,18.98,0.97,0.196,rht502,rht5657)
Al(39.985,18.792,0.97,0.196,rht5657)
0.214211814049 Af() Al(59.188584,47.660896,0.76214
6,0.623574,rht5391) --gt Arb(59.188,47.660,0.762
,0.623,rht5391,rht5653) Al(59.950,48.284,0.762,0
.623,rht5653) 0.214216818695 Rf()
Arb(32.3,58.3,0.77,0.63,rht5183,rht5215) --gt
Arbb(32.3,58.3,0.77,0.63,e5655,rht5183,rht5215)
Rl(32.3,58.3,0.9702,0.196,e5655) 0.214229554433
Af() Al(40.9556,18.988,0.9702,0.196,rht5029)
--gt Arb(40.955,18.98,0.97,0.196,rht502,rht5657)
Al(39.985,18.792,0.97,0.196,rht5657)
17

• SPiM tool is currently being extended with a more
  sophisticated rendering environment that allows
  also 3D plotting.

18
Conclusion

• Our models of actin reflect the interaction of
  individual components with other components. We
  can thus lift the restrictions imposed on
  differential equation models. This results in
  simpler and more accurate models.
• Process algebra techniques bring about a
  flexibility and freedom in construction of
  models, in contrast to differential equation
  models.
• We have introduced new tools and techniques to
  treat process models geometrically, which is
  currently being integrated into the SPiM tool.
• The mechanisms underlying actin polymerisation
  are complex and involve many components.
  Compositional construction of our models
  demonstrate how such components can be included
  at will.
• Our models can be modified and integrated into
  models of other
• systems with the aim of generating hypothesis for
  wet-lab experiments.
• On going work includes integrating an analysis
  on actin-filament length
• and also obtaining models of phagocytosis.