From Red Cells to Skiing to a New Concept for a
Train Track
The American Physical Society
56th Annual Meeting of the Division of Fluid Dynamics
NYC/New Jersey
November 23-25, 2003
Sheldon Weinbaum
Departments of Biomedical &
Mechanical Engineering
New York Center for Biomedical Engineering
The City College of New York, CUNY
Copyright Sheldon Weinbaum, 2003
Collaborators
Red Cells
Hans Vink, Univ. Amsterdam
Jianjun Feng, City College of New York
Mechano-transduction
Xiaobing Zhang, City College of New York
Yuefeng Han, City College of New York
Steve Cowin, City College of New York
Mia Mia Thi, City College of New York
David C. Spray, Albert Einstein Medical College
Skiing and Train track
Qianhong Wu, City College of New York
Yiannis Andreopoulos, City College of New York
Copyright Sheldon Weinbaum, 2003
Red and White Cell Motion in Capillaries
Vink and Duling (1996)
Copyright Sheldon Weinbaum, 2003
Sliding Motion of a Membrane
Over a Thin Surface Glycocalyx
Feng and Weinbaum, JFM 422: 281 (2000)
h2 is fixed in the model.α = h /
2
h1 changes. k=h2/h1
Kp
Copyright Sheldon Weinbaum, 2003
Two-Dimensional Lubrication
Theory for the Brinkman Medium
Brinkman equation:
Dimensionless Reynolds-Type Equation:
Copyright Sheldon Weinbaum, 2003
Pressure Distribution and Equal Pressure
Contours Under a Snowboard
(L/W = 10, h2 = 2cm, α(h2) = 100)
Copyright Sheldon Weinbaum, 2003
Comparison of a Red Cell and SnowBoard
Copyright Sheldon Weinbaum, 2003
Schematic of Dynamic Snow Compression
Apparatus
Copyright Sheldon Weinbaum, 2003
Comparison Between Theoretical and
Experimental Pressure Profiles
Copyright Sheldon Weinbaum, 2003
Periodic Structure of the Endothelial
Glycocalyx
Squire, Chew, Nenji, Neal, Barry and Michel
J. Struct. Biol. 136, 239 (2001)
Copyright Sheldon Weinbaum, 2003
Hexagonal Array seen near Inner Surface
Of Glycocalyx in Freeze-fracture
Squire, Chew, Nenji, Neal, Barry and Michel (2001)
Copyright Sheldon Weinbaum, 2003
Model for Mechanotransduction
Weinbaum et al. Proc. Natl. Acad. Sci. 100, 7988-7996 (2003)
Copyright Sheldon Weinbaum, 2003
Model for Flow in Capillary
Cross-section of capillary
Core protein array
Copyright Sheldon Weinbaum, 2003
Drag Force Distribution on Each Core Protein
5µm diameter capillary
Total drag = 1.4 x 10-3 pN
Copyright Sheldon Weinbaum, 2003
Deflection of Core Protein
Core
Coreprotein
protein
under no
flow
Deformed
core protein
core protein
under flow
Central
axis
Distributed force along core protein
Diagram of loading on core protein
Copyright Sheldon Weinbaum, 2003
Relaxation Of Endothelial Surface Layer
Vink, Duling and Spaan (2001)
Characteristic time
Τ2∗ = 0.38 s
Copyright Sheldon Weinbaum, 2003
Flexural Rigidity of Core Protein
Novel Beam Equation:
c--solid fraction
Characteristic Times:
(short time)
(long time)
Two time constants found by series solution to beam equation
Predicted EI Vink’s Experiment: EI = 700 pN · nm2
Measured EI:
actin (Satcher and Dewey, 1996)
Copyright Sheldon Weinbaum, 2003
distance along fiber (nm)
Deflection of Core Protein
400
300
200
L=400 nm
300
200
150
100
10 dyn/cm2
0
0
5
10
15
lateral deflection (nm)
Copyright Sheldon Weinbaum, 2003
Force Amplification
Copyright Sheldon Weinbaum, 2003
Results: Uniform Laminar flow region
F-actin
Control
DMEM
Thi, Weinbaum and Spray (2003)
τ = 10dyn/cm2
for 5 h with
DMEM,
τ = 10dyn/cm2
for 5 h with
τ = 10dyn/cm2
for 5 h with
DMEM+1% BSA
DMEM+10% FBS
F-actin is redistributed
(more stress fibers throughout cell)
τ = 10dyn/cm2 for 5 h with DMEM
Control (no flow)
275
cellpair1
cellpair2
cellpair3
cellpair4
cellpair5
cellpair6
cellpair7
cellpair8
cellpair9
cellpair10
250
200
150
100
50
225
200
175
150
125
100
75
50
0
5
10
15
20
25
cellpair1
cellpair2
cellpair3
cellpair4
cellpair5
cellpair6
cellpair7
cellpair8
cellpair9
cellpair10
250
Average Density Profile
Average Density Proflie
300
30
0
5
10
15
20
25
30
Cell-Cell Distance (µm)
Cell-Cell distance (µm)
τ = 10dyn/cm2 for 5 h with DMEM + 1% BSA τ = 10dyn/cm2 for 5 h with DMEM + 10% FBS
Average Density Profile
250
225
200
175
150
125
100
300
Average Density Proflie
cellpair1
cellpair2
cellpair3
cellpair4
cellpair5
cellpair6
cellpair7
cellpair8
cellpair9
cellpair10
275
cellpair1
cellpair2
cellpair3
cellpair4
cellpair5
cellpair6
cellpair7
cellpair8
cellpair9
cellpair10
250
200
150
100
75
50
0
5
10
15
20
Cell-Cell Distance (µm)
25
30
50
0
5
10
15
20
Cell-Cell distance (µm)
25
30
Buckling of Initially Curved Beam
compressive force P (pN)
x
δ
δ0
P
y(x)
0.020
y0 (x)=arcsin(y/δ0)/a
y
elastica
δ0=0 nm
0.015
0.010
20
40
60
0.005
0.000
0.15
total
0.025
0
0.12
0.09
small deflection
theory
0.06
0.03
100
200
300
normal deflection (nm)
total compressive force (pN)
Revised Beam Equation
0.00
Pf = 160dyn/cm2
Copyright Sheldon Weinbaum, 2003
ESL Drainage Due to RBC Arrest
L
Lf
Drainage Time
Variable K
Sangani and Acrivos
(1982)
Copyright Sheldon Weinbaum, 2003
ESL Drainage
1.0
maximum compression
Lf/Lf0
0.8
0.6
constant KP
2
PC = 2420 dyn/cm
0.4
variable KP
0.2
0.5
1
2 5
time (s)
25
Copyright Sheldon Weinbaum, 2003
Dynamic Compression with Goose Down
Feasibility of Supporting a Train Car
Pc2=5.2×104Pa Lift force = 260 tons
Copyright Sheldon Weinbaum, 2003
Sketch of the New Train Model in Transverse
Plane
Copyright Sheldon Weinbaum, 2003
Performance of the Enhanced Lift 50 Ton
Train Car
Copyright Sheldon Weinbaum, 2003
Conclusions
! There is a remarkable dynamic similarity between a red cell gliding on
the endothelial glycocalyx and a human skiing though they differ in
size by O(1015).
! For a given planform without lateral leakage lift increases as the
square of the Brinkman permeability parameter α=h/Kp1/2
! For two-dimensional planforms with lateral leakage the lift decreases
as (W/L)2.
! The endothelial glycocalyx is an extraordinary structure whose fibers
are stiff enough to transmit fluid shear stress to the actin cytoskeleton
in initiating intracellular signaling. However, they would easily buckle
during red cell arrest were it not for the fluid draining pressure which
carries most of the normal load.
! The small elastic restoring force of the fibers allows for a huge
reduction in the sliding friction due to the solid phase.
! A highly compressible track with the mechanical properties of goose
down is capable of supporting a 50 ton train car traveling at even
relatively low speeds with minimal sliding friction. At high speeds
there would be little deformation.