Calcium Oscillation in Calcium Oscillation in the Pollen Tube Growththe Pollen Tube Growth

Presented by:Presented by:

XIA,FanXIA,Fan

0305013003050130

OutlineOutline

IntroductionIntroduction SystemSystem Main Players and processMain Players and process

ModelingModeling AssumptionAssumption ModelingModeling Result and discussionResult and discussion

Further workFurther work

IntroductionIntroduction

SystemSystem Pollen tubePollen tube

• Essential for plant Essential for plant reproductionreproduction

• Paradigm for the study Paradigm for the study of polar and of polar and oscillatoryoscillatory growthgrowth

OscillationOscillation• UniversalUniversal• Unclear mechanismUnclear mechanism

IntroductionIntroduction Oscillations of other quantitiesOscillations of other quantities

[Ca[Ca2+2+]c at the tip]c at the tip extracellular Caextracellular Ca2+2+ influx influx F-actin at the tipF-actin at the tip

• NotesNotes• Same periodSame period• Different phaseDifferent phase

CaCa2+2+

o Tip-focused gradientTip-focused gradient• A localized gradient of A localized gradient of

cytosolic free cacytosolic free ca2+2+ at their at their apex. There exist pumps apex. There exist pumps along the shank of the pollen along the shank of the pollen tube.tube.

o Tip-focused extracellular Tip-focused extracellular CaCa2+2+ influx influx• directed inward specifically directed inward specifically

at the tipat the tip• This influx is measured This influx is measured

outside the cell wall. The outside the cell wall. The influx that really entering the influx that really entering the membrane is approximately membrane is approximately 10% of the influx measured.10% of the influx measured.

F-actinF-actino Ring in the subapical regionRing in the subapical region

• In the subapical region, an cortical In the subapical region, an cortical actin collar or ring is observed.actin collar or ring is observed.

o A highly dynamic form of F-A highly dynamic form of F-actin in the extreme apexactin in the extreme apex

• It rapidly appear in and disappear It rapidly appear in and disappear from the cortex of the extreme from the cortex of the extreme apex of pollen tubes, the expected apex of pollen tubes, the expected site of excytosis.site of excytosis.

• The F-actin at the tip refer to this The F-actin at the tip refer to this kind of actin.kind of actin.

o The actin collar in the subapical The actin collar in the subapical region alternates with the highly region alternates with the highly dynamic form in the apex. dynamic form in the apex.

Monteiro et al. 2005

Oscillation of growth rate and tip F-actin Range: 0.1 ~ 0.4 μm sec-1

0°0°F-actin F-actin

at the tipat the tip

180° growth rate

[Ca2+ ]c at the tip

270°270°extracellular extracellular CaCa2+2+ influx influx

There might exists small phase difference between these two.

Holdaway-Clarke et al. 1997

ComplexityComplexity Many playersMany players Many processesMany processes

Gu et al. 2004

Monteiro et al. 2005

Basic ProcessesBasic Processeso ExocytosisExocytosis

• Vesicle docking Vesicle docking and fusionand fusion

o Wall constructionWall constructiono EndocytosisEndocytosis

• Retrieval of the Retrieval of the membrane and membrane and recyclingrecycling

Samaj et al. 2004

Calcium /calcium channelCalcium /calcium channel• [Ca[Ca2+2+]]cc plays an important plays an important

role inrole in dockingdocking and and secretionsecretion of wall materials.of wall materials.

• It helps to release the It helps to release the vesicles from the vesicles from the cytoskeleton by cytoskeleton by depolymerization of F-actindepolymerization of F-actin..

ModelingModeling IdeaIdea

o There exists a microscopic loop.There exists a microscopic loop.o The macroscopic oscillation is the The macroscopic oscillation is the synchronizedsynchronized

behavior of the microscopic events.behavior of the microscopic events.• OscillationOscillation Synchronization Global signal Synchronization Global signal

calcium tip concentrationcalcium tip concentration• Tip concentration can be a good candidates due to fast Tip concentration can be a good candidates due to fast

diffusion.diffusion.

2 2

~

~ / ~ 3 / 0.01 ~ 1

x Dt

t x D ms s

Microscopic storyMicroscopic story

Channel open

[Ca2+]c

increase

F-actin depolymerization

Vesicle docking and fusion

Channel close

[Ca2+]c decrease

Membrane

retrieval Wall

construction growth

AssumptionAssumption

o The open of the channel is a The open of the channel is a stochastic processstochastic process, , the probability of which is negatively correlated the probability of which is negatively correlated to the concentration at the position.to the concentration at the position.• The dependence is not necessarily direct.The dependence is not necessarily direct.• The function form depends on the property of the

channel.• Two possible mechanism: voltage-gated and stretch-

activated.o After open, the channel has to After open, the channel has to restrest for some time for some time

before the next open.before the next open.• The resting time relates to the time for membrane The resting time relates to the time for membrane

retrieval.retrieval.

TargetTarget

o Check the synchronized behavior of the Check the synchronized behavior of the ion channel.ion channel.• Oscillation of tip [Ca2+]c

• Oscillation of tip F-actin

o Other oscillations are not included:Other oscillations are not included:• Oscillation of growth rate• Oscillation of extracellular influx

ParametersParameters

o Physical dimension of the tube Physical dimension of the tube 33μμm,3m,3μμm,40 m,40 μμmmo Diffusion constant Diffusion constant 0.01 0.01 μμmm2 2 msms-1-1

o Influx of one channel Influx of one channel 5*10-9 pmole ms-1(which corresponding to 500 ions each ms)

o Channel density 100μμmm-2-2

• This data is the channel density in the neuron.This data is the channel density in the neuron.• The extracellular Ca2+ influx is around 2*10The extracellular Ca2+ influx is around 2*10-11-11pmolepmoleμμmm-2-2 ms ms-1-1

• The influx of each channel is around 5*10The influx of each channel is around 5*10-9-9pmole mspmole ms-1-1

• Assume the open of the channel behaves like the firing of neuron cells, Assume the open of the channel behaves like the firing of neuron cells, i.e. open for 1ms and then close.i.e. open for 1ms and then close.

• From the above assumptions, we can calculate the resting time.From the above assumptions, we can calculate the resting time.o Resting time Resting time 25,000ms25,000ms 9

11

5*10 *10025000

2*10ms

ModelingModeling

Probability FunctionProbability Functiono We use the tanh

function for voltage-gated channel.

ModelingModeling

Sink: Corresponding to the pump in the shank

and the X sink in the cytosol.

diffusioninflux

periodic boundary condition

channels

ModelingModeling

Modeling of the whole pollen tubeModeling of the whole pollen tubeo Too heavy computation loado Modification: considering the channel area

only

ModelingModeling

Lattice model of the Lattice model of the channel area onlychannel area onlyo ParameterParameter

• Physical dimension: 3Physical dimension: 3μμm,3m,3μμm m • The others remain the sameThe others remain the same

o Free space approximationFree space approximation

2

3

( )2exp( )

44j i

ji ii

r rmC

DtDt

i3i3

i2i2jj

i1i1

channels

m: amount of Ca deposit by one firing

D: diffusion constant

r: position of the channel

t: duration after the open of the channel

ModelingModeling

o How good is the free space approximationHow good is the free space approximation

3um x 3um

Open one channel and examine the concentration changes of the three channels in different models: pollen tube with sink, pollen tube without sink and free space.

ModelingModeling

ModelingModeling

ModelingModeling

Conclusion:The free space approximation is not bad.

ModelingModeling

o To reduce the To reduce the computation loadcomputation load• Remove the Remove the

channels’ effect channels’ effect after 300ms from its after 300ms from its open time.open time.

ResultResult

DiscussionDiscussion

DiscussionDiscussiono Synchronized behavior and oscillation of Synchronized behavior and oscillation of

[Ca[Ca2+2+]]tiptip

o The period of oscillation is determined by The period of oscillation is determined by channel density/ resting time.channel density/ resting time.

o The synchronized behavior results from the The synchronized behavior results from the same/little spread resting time.same/little spread resting time.

o Biological support is needed.Biological support is needed.

DiscussionDiscussion

Shortcomings:Shortcomings:• Parameters and Parameters and

mechanisms are mechanisms are borrowed from other borrowed from other system.system.

• Steady stateSteady state

Mathematical AnalysisMathematical Analysis

Equation:Equation:

Steady StateSteady State

1( ) ( )

1( ) ( )

BB B

R

BD

dnn N n P C

dt

dCC m N n P C

dt

nB: number of channels in blocked stateC: concentration in the tip region

N: total number of channelsTR: resting timeTD: diffuse time m: concentration increase due to one channel

* * *

* * *

1( ) ( ) 0

1( ) ( ) 0

B BR

BD

n N n P C

C m N n P C

* * *

* * *

1( ) ( ) 0

1( ) ( ) 0

B BR

BD

n N n P C

C m N n P C

(C* ,nB*) :steady state solution

Stability AnalysisStability Analysis

Small Perturbation:Small Perturbation:*

* * * ' *

* * ' *

( ) 1( ) ( )( ( ) ( ) )

( ) 1( ) ( )( ( ) ( ) )

BB B

R

BD

d n nn n N n n P C P C C

dt

d C CC C m N n n P C P C C

dt

* * ' *

* * ' *

( ) 1( ( )) ( ) ( )

( ) 1( ) ( ( ) ( ))

BR

BD

d nP C n N n P C C

dt

d CmP C n m N n P C C

dt

* * ' *

'

'* * ' *

1( ( )) ( ) ( )

1( ) ( ) ( )

BR

BD

P C N n P Cn n

CC mP C m N n P C

simplification

Matrix form

Stability AnalysisStability Analysis

* * ' * * ' * *1 1( ( ))( ( ) ( )) ( ) ( ) ( ) 0B B

R D

P C m N n P C N n P C mP C

Eigenfunction

2

* * ' *

* ' * *

0

1 1( ( )) ( ( ) ( ))

1 1 1 1( ( ) ( )) ( )

BR D

BR D R D

b c

b P C m N n P C

c m N n P C P C

b>0 & c>0

stable

Phase DiagramPhase Diagram

Further WorkFurther Work

More reliable parametersMore reliable parameters More biological informationMore biological information New variablesNew variables

Resting time (function of concentration)Resting time (function of concentration) Stretch (stretch-activated)Stretch (stretch-activated)

AcknowledgementAcknowledgement

Supervisor: Supervisor: Prof. L.H. TangProf. L.H. Tang Observer: Observer: Prof. N.H. CheungProf. N.H. Cheung Group Members: Group Members: Tony,Tony, Chunhui CaiChunhui Cai, ,

Chao WangChao Wang, , Zhu YangZhu Yang..