Model-Convolution Approach to Modeling Green Fluorescent

Protein Dynamics: Application to Yeast Cell Division

David Odde

Dept. of Biomedical Engineering

University of Minnesota

Mitotic Spindle

spindle pole

chromosomes

kinetochore

1.7 µmIn budding yeast:

~40 MTs10-20 µm

In animal cells:

~1000 MTs

interpolarmicrotubule

- -

++

++

kinetochore microtubule

bifunctionalplus-end motors

+ +

spindle pole

COMPRESSION

TENSION

Microtubule Dynamic Instability

Leng

th (

µm

)

Time (minutes)

“Catastrophe”

“Rescue”

Microtubule “Dynamic Instability”

Vg

Vs

kc

kr

Hypothesis: The kinetochore modulates the DI parameters

Can only get peaks here

Not here

MT Length Distribution for Pure Dynamic Instability

Right PoleLeft Pole

1.7

Budding Yeast Spindle Geometry

Congression in S. cerevisiae

P PEQ

Green=Cse4-GFP kMT Plus Ends

Red=Spc29-CFP kMT Minus Ends

“Experiment-Deconvolution”vs. “Model-Convolution”

Model Experiment

Deconvolution

Convolution

Point Spread Function (PSF)

• A point source of light is spread via diffraction through a circular aperture

• Modeling needs to account for PSF

-0.4-0.20+0.2+0.4 μm

Simulated Image Obtainedby Model-Convolution of

Original Distribution

Original FluorophoreDistribution

Image Obtained by Deconvolution

of Simulated Image

Potential Pitfalls of Deconvolution

Cse4-GFP Fluorescence Distribution

Experimentally Observed

Theoretically Predicted

Dynamic Instability Only Model

Sprague et al., Biophysical J., 2003

Modeling ApproachModel

Probability that themodel is consistent with the data

ParameterSpace

(a1, a2, a3,…aN)

<Cutoff?

Experimental Data yes

no

Accept Model

ParameterSpace

Reject Model

ParameterSpace

Accept Model

ParameterSpace

Modeling ApproachModel assumptions:1) Metaphase kinetochore microtubule dynamics

are at steady-state (not time-dependent)2) One microtubule per kinetochore3) Microtubules never detach from kinetochores4) Parameters can be:• Constant• Spatially-dependent (relative to poles)• Spatially-dependent (relative to sister

kinetochore)

“Microtubule Chemotaxis” in a Chemical Gradient

ImmobileKinase

MobilePhosphatase

A: Phosphorylated ProteinB: Dephosphorylated Protein

k*Surface reaction B-->A

kHomogeneous reaction A-->B

KinetochoreMicrotubules

- +

ImmobileKinase

MT Destabilizer

Position

Concentration

X=0 X=L

Could tension stabilize kinetochore microtubules?

Tension

Kip3

Distribution of Cse4-GFP: Catastophe Gradient with Tension Between Sister Kinetochore-Dependent Rescue

Model Combinations

123

Catastrophe Gradient-Tension Rescue Model

Conclusions

• Congression in budding yeast is mediated by:– Spatially-dependent catastrophe

gradient– Tension between sister kinetochore-

dependent rescue

• Model-convolution can be a useful tool for comparing fluorescent microscopy data to model predictions

Acknowledgements

• Melissa Gardner, Brian Sprague (Uof M)

• Chad Pearson, Paul Maddox, Kerry Bloom,Ted Salmon (UNC-CH)

• National Science Foundation

• Whitaker Foundation

• McKnight Foundation

Simulated Image Obtainedby Convolution of PSF and GWN

with Original Distribution

Original FluorophoreDistribution

Model-Convolution

Kinetochore MT Lengths in Budding Yeast

Experimentally Observed

Theoretically Predicted

?

2 µm

Catastrophe Gradient Model

Fre

quen

cy (

min

-1)

Normalized Spindle Position

Sprague et al., Biophys. J., 2003

Distribution of Cse4-GFP: Catastrophe Gradient Model

Experimental Cse4-GFP FRAP

•Cse4-GFP does not turnover on kinetochore

•Kinetochores rarely persist in opposite half-spindle

Pearson et al., Current Biology, in press

Cse4-GFP FRAP: Modeling and Experiment

Catastrophe Gradient Simulation

Experiment

Cse4-GFP FRAP: Modeling and Experiment

Gradients in Phospho-state

1.0

0.8

0.6

0.4

0.2

0.0

Con

cen

trati

on

, Y

1.00.80.60.40.20.0

Position, X

If k= 50 s-1, D=5 µm2/s, and L=1 µm, then =3

MT Destabilizer

Position

Concentration

X=0 X=L

Could tension stabilize kinetochore microtubules?

TensionTension

Kip3

Catastophe Gradient with Tension Between Sister Kinetochore-Dependent Rescue Model

Experimental Cse4-GFP in Cdc6 mutants

WT Cdc6

Cse4-GFP in Cdc6 Cells: No tension between sister kinetochores

Rescue Gradient with Tension-Dependent Catastrophe Model (No Tension)

Normalized Spindle Position

Fre

quen

cy (

min

-1)

Catastrophe Gradient with Tension-Dependent Rescue Model (No Tension)

Fre

quen

cy (

min

-1)

Normalized Spindle Position

Cse4-GFP in Cdc6 Cells: No tension between sister kinetochores

0.022

0.023

0.024

0.025

0.026

0.027

0.028

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Spindle Position

Frac

tion

Fluo

resc

ence

Experimental cdc6 mutants- No Replication (n=27)Catastrophe Gradient with Tension-Dep. Rescue (No Tension); p=0.11Rescue Gradient with Tension-Dep. Catastrophe (No Tension); p<<.01

Rescue Gradient Model

Normalized Spindle Position

Ca

tast

rop

he

or

Re

scu

e F

requ

enc

y (m

in-1)

Simulation of Budding Yeast Mitosis

Metaphase AnaphasePrometaphase

Start with randompositions, let simulationreach steady-state

Eliminate cohesion,set spring constant to 0

MINIMUM ABSOLUTE SISTER KINETOCHORE SEPARATION DISTANCE

WT Stu2p-depleted

Pearson et al., Mol. Biol. Cell, 2003

Stu2p-mediated catastrophe gradient?

Green Fluorescent Protein

M

D

Prometaphase Spindles and the Importance of Tension in Mitosis

“Syntely”

Ipl1-mediated detachment of kinetochores under low tension

Dewar et al., Nature 2004

MT Length Distributions•Regard MT dynamic instability as diffusion + drift•The drift velocity is a constant given by

•For constant Vg, Vs, kc, and kr, the length distribution is exponential

p x ~ eVdDx

Vd<0 exponential decayVd>0 exponential growth

Vd x Lg Lstc

Vg tg Vststg ts

Vgkc

Vs kr1kc

1kr

Sister Kinetochore Microtubule Dynamics

Simulated Image Obtainedby Convolution of PSF and GWN

with Original Distribution

Original FluorophoreDistribution

Model-Convolution

“Directional Instability”

Skibbens et al., JCB 1993

Tension on the kinetochore promotes switching to the growth state?

Skibbens and Salmon, Exp. Cell Res., 1997

Tension Between Sister Kinetochore-Dependent Rescue

kr kroeF

Catastrophe Gradient withTension-Rescue Model

Lack of Equator Crossing in the CatastropheGradient with Tension-Rescue Model

~25% FRAP recovery ~5% FRAP recovery

Microtubule Dynamic Instability

Model for Chemotactic Gradients of Phosphoprotein State

cAt

D 2cAx2

kcA Fick’s Second Law with First-Order HomogeneousReaction (A->B)

DcAx x0

k *cB 0 B.C. 1: Surface reaction at x=0 (B->A)

DcAx xL

0 B.C. 2: No net flux at x=L

cA cB cT Conservation of phosphoprotein

Sprague et al., Biophys. J., 2003

Predicted Concentration Profile

where

Y cA cT

X x L

kL2

D

A*e2

e2 1 * 1 e2 B*

e2 1 * 1 e2 * k

*LD

Y Ae X BeX

Model Predictions: Effect of Surface Reaction Rate

1.0

0.8

0.6

0.4

0.2

0.0

Con

cen

trati

on

, Y

1.00.80.60.40.20.0

Position, X

Defining “Metaphase” in Budding Yeast