Project C3: Optical Stretcher Model for Membrane Mechanics
David W.M. Marr: Colorado School of Mines
Charles Eggleton: University of Maryland, Baltimore County
Specific Objectives: The specific objective of the collaboration is to develop and apply a
computational model of capsule deformation1 to predict droplet, vesicle, and biological cell
response under combined optical deformation and hydrodynamic shear conditions as a function
of the mechanical model of the membrane.
Background/Rationale: With the experimental data we will obtain during Core 3 development,
Dr. Eggleton will be able to greatly enhance and validate his theoretical models. Experimental
observation of cell deformation in micro-channels will be analyzed by using the mathematical
model based on the immersed boundary method2 to yield estimates of the mechanical
properties of the cell membrane. These simulations will help calculate important biophysical
parameters such as the dilatational and extensional (shear) moduli and Poission’s ratio of the
cell membrane. The collaborator and coworkers have investigated the effect of shear on the
cell-substrate contact area3. These simulations show that the contact area increases with shear
rate for the more compliant membrane whereas no appreciable effects are observed in the cell
with membrane stiffness of 3.0 dyn/cm (relatively stiff). The contact area estimates for cells with
membrane stiffness of 0.3 dyn/cm compare well with those observed during in vivo leukocyte
rolling where the contact are increased from 19.6 to 35.8 square microns when the shear rate
was increased from 100 to 400 s-1.
Design/Methods: The transient axisymmetric deformation of an interface in an unbounded fluid
can be calculated using the Boundary Integral Method (BIM). We have used the BIM to simulate
the effects of a surfactant monolayer on interfacial dynamics 4-6. In this, the velocity of point xo
on a cellular membrane or drop (particle) interface that is suspended in and encloses a fluid
interface, as shown in Figure 1, can be represented in the form7, 8,
u j xo  
2 
1
1 
u j xo  
f x Gij x, xo dS x  
ui x Tijk x, xo nk x dS x 
1 
41 1   
4 1   


s
S
where the integration is over the total interfacial area S. The free-space velocity Green's
function tensor G and the associate stress tensor T, are called Stokeslet and stresslet,
respectively. The function  is the jump in the traction across the interface. For the optical
stretcher,  is the sum of the tractions exerted by the incident beam and the tractions exerted
by the cell membrane due to deformation. Note that the effects of a known external flow u j can
be modeled and a quiescent flow can be considered by letting u j equal zero everywhere. An
important feature of the mathematical model is that the velocity at a given time instant depends
only on the position of the interfaces at this time instant. Given the position of the interface at a
given time instant, the jump in the traction across the drop’s interface  is found from the
expressions for the force from an incident light beam, and the membrane constitutive equation
or equation of state.
Optical tractions on an interface: The components of the traction force on the interface were
developed by Guck et al9. The traction force exerted on the interface is due to the change in
momentum between the incident light ray and the reflected and transmitted rays. The magnitude
of the traction force exerted on the interface or cell membrane is given by:
F
n1QP
c
where n1 is the refractive index, Q is a factor that describes the amount of momentum
transferred (Q= 2 for reflection, Q =1 for absorption), P is the incident light power or intensity,
and c is the speed of light. The components of the traction force are given in terms of Q on the
front side, parallel and perpendicular to the beam axis, are
perpendicular
parallel
( )  Q front ( ) sin 
Q front
( )  Q front( ) cos  and Q front
where  is the angle between the beam axis and the direction of the momentum transferred,
and is the angle between the incident ray and the local interface. Similarly, on the back side of
the particle, the components of the surface force are
parallel
perpendicular
Qback
( )  Q front( ) cos  and Qback
( )  Q front( ) sin 
The components of the optical traction force parallel and perpendicular to the incident beam will
be transformed to components normal and tangential to the local interface. The optical forces
will be incorporated into the boundary integral code and will be used to calculate the net traction
force exerted on the interface.
Membrane Constitutive Laws: For the elastic membrane capsules being considered, the traction
jump is caused by the elastic tensions in the membrane. For an axisymmetric domain the
traction force  depends on the elastic tensions and the curvature of the interface and is given
by7, and is given by




  
1 r

f   s s     n   s 
 s    t
 s r s

where s and  are the principal curvatures and s and  are the tension components in the
meridional and azimuthal principal directions, respectively. These parameters are dependent
upon the constitutive equation used. The unit normal vector n points in the direction of the
external fluid and is normal to S, and the unit vector t is tangential to the interface along the
meridional plane as shown in Figure 1. In an elastic membrane, tensions arise due to the
deformation of the material. Tensions depend on the elastic parameters and in an axisymmetric
domain are functions of the principal extension ratios
s 
ds
dso
and r 
r
ro
where so and ro are the local arc length and radial
position of the material element in the undeformed
state. A variety of two-dimensional constitutive
equations are used to model elastic membranes,
such as the two-dimensional Hooke’s law for linearly
elastic material, Mooney-Rivlin law for a very thin
isotropic volume-incompressible membrane, and
others. In this study we will plan to use the Evans
and Skalak law developed to represent the
erythrocyte membrane.
Evans and Skalak law: The strains within a
membrane can be separated into two components: a
strain deviator which measures the change in shape
of an element at constant area and a hydrostatic
strain which measures the change in area or
dilatation _A/A of the membrane. Materials may
behave differently when individually subjected to
Figure 1: Meridian of the trace of an
axisymmetric particle with polar axis z. The
drop is suspended in a fluid of viscosity , the
internal viscosity is . Deformation is defined
in terms of the minor and major diameters DF
= (A-B)=(A+B). The interface is defined in
terms of the arclength parameter s measured
from the pole.
each type of strain. It was observed that a red blood cell membrane changes shape (shears)
very easily while strongly resisting relative area changes. For this reason, Evans and Skalak10
defined two linearly independent invariants,
 ES  s  1 and  


1
2s  2  1 .
2s 
The invariant  is the fractional change in the area of the surface element and  measures the
extension of the element at constant element area. For an Evans and Skalak (ES) material the
principal tension components are given by
 s  K ES ES 
 
G ES
2 2s 2
2
s
 2
 and   K
ES ES

    
G ES
2 2s 2
2
2
s
The shear modulus GES and the dilation modulus KES determine the amount of energy required
to deform and change the area of the material, respectively. The parameters KES and GES are
independent and attributed to the properties of the lipid bilayer and cytoskeleton, respectively.
Membrane Mechanics and Rheology: The microfluidic device can be used to transport and
position cells within the optical stretcher test section. Deformation of an erythrocyte under one
or two incident beams and relaxation to an undeformed state will be simulated and observed.
Comparison of the simulations with the observations will be used to determine the elastic moduli
KES and GES, and internal viscosity of the cell. Simulations will be conducted for increasing
incident beam intensities for the constitutive models described above. This will form a data base
of simulations that can be used for comparison with observations. The elastic moduli obtained
through curve fitting will depend on the constitutive equation chosen to model the membrane
mechanics. Alternative constitutive equations can be chosen or developed when significant
differences are seen between the observations and the simulations. The cell membrane
rheometer based on the optical stretcher will be developed, calibrated and validated by
observing the stretching and relaxation of erythrocytes and comparing with simulations using
the Evans and Skalak model.
Impact: To interpret experimental data, modeling of cell deformation under forces due to shear
and optical manipulation is critical for the successful application of the technologies being
developed under Core 3.
References
1.
Eggleton, C. D. & Popel, A. S. Large Deformation of Red Blood Cell Ghosts in a Simple
Shear Flow. Physics of Fluids 10, 18341845 (1998).
2.
Peskin, C. & McQueen, D. A Three-Dimensional Computational Method for Blood Flow
in the Heart I. Immersed Elastic Fibers in a Viscous Incompressible Fluid. Journal of
Computational Physics 81, 372 (1989).
3.
Jadhav, S., Eggleton, C. D. & Konstantopoulos, K. Computational Model Predict that
Cell Deformation Affects Selectin-Mediated Leukocyte Rolling. Biophys J 88, 96-104
(2005).
4.
Eggleton, C. D., Pawar, Y. P. & Stebe, K. J. Insoluble Surfactants on a Drop in an
Extensional Flow: A Generalization of the Stagnated Surface Limit to Deforming
Interfaces. Journal of Fluid Mechanics 385, 79-99 (1999).
5.
Eggleton, C. D. & Stebe, K. J. An Adsorption-Desorption Controlled Surfactant on a
Deforming Droplet. Journal of Colloid and Interface Science 208, 68-80 (1998).
6.
Eggleton, C. D., Tsai, T. M. & Stebe, K. J. Tip Streaming from a Drop in an Extensional
Flow in the Presence of Surfactants. Physical Review Letters 87, 048302 (2001).
7.
Pozrikidis, C. Interfacial Dynamics for Stokes Flow. Journal of Computational Physics
169, 250-301 (2001).
8.
9.
10.
Pozrkidis, C. Boundary Integral and Singularity Mehods for Linearized Viscous Flow
(Cambridge University Press, 1992).
Guck, J., Ananthakrishnan, R., Moon, T. J., Cunningham, C. C. & Käs, J. Optical
Deformability of Soft Biological Dielectrics. Physical Review Letters 84, 5451-5454
(2000).
Evans, E. A. & Skalak, R. Mechanics and Thermodynamics of Biomembranes (CRC
Press, Boca Raton, Florida, 1980).