SLIDE 1 Driving Forces for Cell Cluster Shape Evolution and Stability
- Acknowledgment. This work is supported primarily by the MRSEC Program of the National
Science Foundation at Brown University under award DMR-0520651.
Asha Nurse, L.B. Freund and J. Youssef School of Engineering, Brown University Applied and Computational Mathematics Seminar Series 04/19/2011
SLIDE 2
– Experimental Background
– Origin of Surface Energy – Analysis of the Model – Results & Conclusions
- Linear Stability of a Toroidal Cluster
– Stability Analysis – Formulation of Model
- Calculating Surface Area and Volume of Perturbed shape
- Alternate Perturbed Shape
– Non-constant Surface Energy Density – Results & Conclusions
Outline
SLIDE 3
Dean, D. M., Napolitano, A. P., Youssef, J. and Morgan, J. R. FASEB 2007
Directed Self Assembly Experiments
SLIDE 4 The Phenomenon
800 μm cell suspension polyacrylimide chamber solvent toroidal cluster
A mathematical model of the current system will
- 1. allow for understanding the factors involved in cell
cluster reorganization
- 2. offer a framework to organize lab observations
SLIDE 5 Spontaneous Climb
200μm
2 hrs 4 hrs 6 hrs
SLIDE 6
The Model
Nurse, A., Youssef, J., Freund, L.B. JAM 2011 Under Review
SLIDE 7
Possible Origin of Surface Energy
Higher free energy tension bundles
SLIDE 8
The Model
SLIDE 9
Finding the Surface Flux
SLIDE 10
A Variational Approach
SLIDE 11
κ = 15 20 25
β(τ) a(τ)/b0 β(τ) α(τ)/b0 κ = 25 ω =100
τ Numerical Solution for Shape Evolution
κ = κeq
SLIDE 12
β(τ) α(τ)/b0 κ = 25 ω =100
τ
κ = 15 20 25
β(τ) a(τ)/b0
Numerical Solution for Shape Evolution
κ = κeq
SLIDE 13
α = 85 65 55
a(τ)/b0 β(τ)
τ Numerical Solution for shape evolution
SLIDE 14
Extraction of Surface Energy Density Values
time (hrs) γ (mJ/m2) ms 10-14 (m4hr/g) 2 0.0580 0.543 4 0.0638 0.920 6 0.0667 1.238 8 0.0693 1.386 α1= 55°, α2 = 65°
Foty & Steinberg (05) & Sivansankar et al (99): 1.556 x 10-4 ≤ γ ≤ 1.522 x 10-3 mJ/m2
SLIDE 15
Conclusions from the Basic Model
SLIDE 16 Outline
– Experimental Background – Origin of Surface Energy – Riemannian Geometry – Analysis of the Model – Results & Conclusions
- Linear Stability of a Toroidal Cluster
– Stability Analysis – Formulation of Model
- Calculating Surface Area of Perturbed shape
- Alternate Perturbed Shape
– Non-constant Surface Energy Density – Results & Conclusions
SLIDE 17
Stability of Toroidal Shapes
McGraw et al, Soft Matter 2010 Pairam & Fernandez-Nieves, Physics Rev. Letter 2009 a0/ b0 ≈ 1.9 a0/ b0 ≈ 8.7 Jeff Morgan Lab
SLIDE 18
Linear Stability Analysis of the Toroidal Cluster
SLIDE 19
Linear Stability Analysis of the Toroidal Cluster
SLIDE 20
γ is spatially constant
Always stable
SLIDE 21
Alternate perturbed shape
Inconclusive
SLIDE 22
a0 r0
γ is not spatially constant
a0 r0
θ
SLIDE 23
r0/b0 = 4
stable unstable
Nonuniform γ with same applied perturbation
SLIDE 24
r0/b0 = 4
stable unstable
Non uniform γ with same applied perturbation
r0/b0 = 8
stable unstable
SLIDE 25
γ is a function of position on the surface
SLIDE 26
Conclusions on Toroidal Stability
SLIDE 27 Summary of Results on Toroidal Clusters
Formation of uniform toroidal self-assembled cluster at base of chamber Cluster spontaneously climbs conical pillar to reduce its surface area Remains at base of chamber undergoes localized thinning around circumference For family of applied perturbation
- γ must be a variable
- γ (θ) sufficient for instability
- r0/b0 affects cluster stability
Rate of climb affected by
- influence of SE/GPE (κ)
- slope of conical pillar (α)
- surface mobility of diffusing cells
SLIDE 28
- Origin of γ requires further investigation
- More precise method required to extract values of ms
- Conduct linear stability analysis of toroidal shapes with
– other family of perturbations – more complex surface energy density function
- For spatially constant γ, conduct a nonlinear stability analysis of the
toroidal clusters
Further Work and Questions
SLIDE 29 Acknowledgements & Publications
200 μm
SLIDE 30
