Lecture 7: Multiscale Bio-Modeling and
Visualization
Cell Structures: Membrane and Intra-Cellular
Molecule Models (NMJ)
Chandrajit
Bajaj
http://www.cs.utexas.edu/~bajaj
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Molecules of the Cell
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Bacterial Cell
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Functions performed by Cells
• Chemical – e.g. manufacturing of
proteins
• Information Processing – e.g. cell
recognition of friend or foe
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Neuromuscular Junction (NMJ)
Movie!
http://fig.cox.miami.edu/~cmallery/150/neuro/neuromuscular-sml.jpg
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Cells of the Central Nervous System
Figure 8-3 Anatomic and functional categories of neurons
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
How do Nerve Cells Function ?
Center for Computational Visualization
Institute of Computational and Engineering Sciences
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University of Texas at Austin
September 2005
Axonal transport of membranous organelles
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Synapse
• Dendrite receives signals
• Terminal buttons release
neurotransmitter
• Terminal button pre-synaptic
• Dendrite post synaptic
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Membrane Proteins
• Ligand Gated channels bind
neurotransmitters
• Voltage gated channels propagate
action potential along the axon
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Neurotransmitters
• Released from the terminal
buttons
• Bind to ligand gated receptors on
the post-synaptic membrane
• Can excite or repress electrical
activity in neuron
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Electrical Excitation
• Excitatory neurotransmitters in
brain such as Glutamate released
from terminal button, bind ligand
gated post synaptic ionotrophic
membrane proteins
• Opens Ca+ channels and excites
the neuron
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
All or None
• If threshold
potential
reached, the axon
hillock generates
an action
potential
• Voltage dependent
Na and K channels
propagate along
the axon
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Propagation of an action potential along an axon without attenuation
Action potentials are the direct consequence of the properties of
voltage-gated cation channels
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Action Potential I
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Action Potential II
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Propagation in Axons
• The narrow cross-section of axons and
dendrites lessens the metabolic expense
of carrying action potentials
• Many neurons have insulating sheaths of
myelin around their axons. The sheaths
are formed by glial cells.
• The sheath enables the action
potentials to travel faster than in
unmyelinated axons of the same diameter
whilst simultaneously preventing short
circuits amongst intersecting neurons.
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Terminal Buttons
• Electrical excitation signals the
release of neurotransmitters at
terminal button
• Neurotransmitters stored in fused
vesicles
• Release at pre-synaptic membrane
by exocytosis
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Chemical synapses can be excitatory or inhibitory
Excitatory neurotransmitters open cation channels, causing an influx of Na+ that depolarizes the
postsynaptic membrane toward the threshold potential for firing an action potential.
Inhibitory neurotransmitters open either Cl- channels or K+ channels, and this suppresses firing by
making it harder for excitatory influences to depolarize the postsynaptic membrane.
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Neuromuscular Junction (NMJ)
Movie!
http://fig.cox.miami.edu/~cmallery/150/neuro/neuromuscular-sml.jpg
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
How do Synapses Occur at the Neuro-Muscular Junction ?
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Biological / Modeling Motivation - NMJ
• Complex model with intricate geometry,
intriguing physiology and numerous
applications
• Many diseases/disorders can be traced
back to problems in the Synaptic well
– Myasthenia Gravis: muscle weakness
– Snake venom toxins: block synaptic
transmission
• Holds the key to understanding numerous
biological processes
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Populating the Domain with ≈ 1 million molecules
Image from : www.mcell.cnl.salk.edu[5]
Center for Computational Visualization
Institute of Computational and Engineering Sciences
September 2005
Department of Computer Sciences
University of Texas at Austin
NMJ Multi-Scale Modeling
• Length Scale
– The cell membranes are ≈ Microns
– The receptor molecules are ≈ nanometers
– The ions are ≈ Angstroms
– The packing density is non-uniform
• Time Scale
– The Neurotransmitters diffuse in
microseconds
– The Ion channels open in milliseconds
– The ACh hydrolyzation is in microseconds
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Extracting Domain Information from Imaging
data
• Cellular Membrane Geometry can be extracted (metaballs)
• Receptors are concentrated in certain areas along
the pots-synaptic membrane
• Acetyl-Cholinesterase exists in clusters in the
synaptic cleft
Images from : www.starklab.slu.edu
Center for Computational Visualization
Institute of Computational and Engineering Sciences
September 2005
Department of Computer Sciences
University of Texas at Austin
Synaptic Cleft Geometry
Twin resolution models for the Ce
From 14813 vertices and 29622 triangles to 4825 vertices and 9636 triangles
(~67% decimation)
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Acetylcholinesterase in Synaptic
Cleft
• Activity Sites
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Activity Sites
Cell Membrane
Enlarged View
AchE molecule
(PDBID = 1C2B)
Datasets from www.pdb.org and Dr. Bakers group
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Nictonic-Acetyl-Choline Receptor
Pentameric Symmetry in AchR molecule (PDBID: 2BG9)
[8]
Image from Unwin
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
AChBP (1I9B.pdb) Active Sites
Complementary
component
ACh Binding Site
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
Primary
Component
September 2005
Specificity
• Ion channels are highly specific
filters, allowing only desired
ions through the cell membrane.
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Populating the Membrane with the
molecules
Name
PDBID
Size (oA)
Weight
(kDa)
Density
(/µm2)
NumberAtoms
AChE
1C2B
(58, 65,
58)
160
600 2500
4172
AChR
2BG9
(84, 85,
162)
290
2500 10000
14929
ACh
1AKG
(13, 22,
13)
13.4
30000 - 18
50000
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
RBF Spline Representations of 3D Maps
Local maxima and minima of
the original density map
Original Map
Thin-plate spline interpolation with centers at local max & min
1139 centers, 9.55% error (middle); 7649 centers, 7.36% error (right)
RBF Approximation (5891 centers, 7.88% error)
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Fast and Stable Computation of RBF
Representation of 3D Maps
•
Interpolate Map with an analytic basis of the form
M
s( x)  p( x)   i ( x, xi )
•
i 1
p = polynomial of degree k-1
•  ( x, x ) = Radial basis function (thin-plate spline kernel)
i
•
Make Coefficients
i orthogonal to polynomials of degree k-1
M
  q( x )  0
i 1
i
i
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Thin-Plate Spline Kernel
• One choice for
 
( x1 , x2 )
 (r )  r 2 log r
 
r || x1  x2 ||2
It minimizes “bending energy”:


I  f    f xx2  f xy2  f yy2 dxdy
R2
It is conditional positive definite
• Memory storage
O( N 2 )
• Computational cost
O( N 3 )
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Matrix Form
s  A
0    P T
  
P    ~ ~
 A



0 c 
A ~ non  positivedefinite
~
A ~ positivedefinite
si  function value at xi
Pij  pi ( x j ) , where pi(x) forms a basis for polynomial of degree k-1
i  coefficients of the RBF kernel at xi
 ( x1 , x1 ) ( x1 , x2 )
 ( x , x ) ( x , x )
2
1
2
2
A




( xM , x1 ) ( xM , x2 )
( x1 , xM ) 
 ( x2 , xM ) 




  ( xM , xM ) 

Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Poor Conditioning
Matrix A (1065x1065)
Condition number = 2.95E+06
(non-positive definite)
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Use of
Pre-conditioners/Sparsifiers
Matrix A (1065x1065)
Condition number = 2.95E+06
(non-positive definite)
Multi Scale matrix after HB wavelet
pre-conditioning/sparsification
Condition number = 332
(positive definite)
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Synaptic Cleft Modeling
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
NMJ – Physiology: Synaptic Transmission
Ach = AcetylCholine, AchE = AcetyleCholinEsetrase,
AchR = AcetylCholineReceptor
Image from : Smart and McCammon[1]
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Modeling Physiology I :Electrostatics
Potential
2
à r á[" ( r r V( r )] + k ( r )sinh( V( r )) = ú( r )
Poisson-Boltzmann
" (r )
k2
ú( r )
dielectric properties of the solute and solvent,
ionic strength of the solution ,
solute atomic partial charges
.
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Fas2 meets AChE
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Adaptive Boundary Interior-Exterior Meshes
(a) monomer mAChE
(d) exterior meshes
(b) cavity
(c) interior mesh
•Y. Zhang, C. Bajaj, B.
Sohn, Special issue of
Computer Methods in
Applied Mechanics and
Engineering (CMAME) on
Unstructured Mesh
Generation, 2004.
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Center for Computational Visualization
AChE Tetramer Conformations
Institute of Computational and Engineering Sciences
September 2005
Department of Computer Sciences
University of Texas at Austin
Model Physiology II
Reaction Diffusion Models
• Time dependent equations to model the
diffusion of ACh across the synaptic
cleft
Boundary Conditions
On the domain
at the AchR boundaries
Initial Condition
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Steady State Smulochowski Equation
(Diffusion of multiple particles in a potential field)
 




Jp ( r )  D ( r )[ p ( r )  p ( r )U ( r )]
•  -- entire domain
•  -- biomolecular domain
•  -- free space in 
• a – reactive region
• r – reflective region


p( r )  pbulk for r  b


p( r )  0 (Dirichlet BC) for r  a
  
 
or n  Jp( r )   ( r ) p( r ) (Robin BC)
  
n  Jp( r )  0 for x  r
• b – boundary for 
Diffusion-influenced biomolecular
reaction rate constant :

k
a
  
n  Jp( r )dS
pbulk
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Active Sites of AChE
4.50E+012
1C2O
1C2B
Int2
Monomer*2
Monomer*3
Monomer*4
4.00E+012
3.00E+012
-1
Rate (M min )
3.50E+012
-1
2.50E+012
2.00E+012
1.50E+012
1.00E+012
5.00E+011
0.00E+000
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
I (M)
•Y. Song, Y. Zhang, C. Bajaj, N. A. Baker, Biophysical Journal, Volume 87, 2004, Pages 1-9
•Y. Song, Y. Zhang, T. Shen, C. Bajaj, J. A. McCammon and N. A. Baker, Biophysical Journal, 86(4):2017-2029, 2004
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
Many Next Steps
• Poisson-Boltzmann equation for electrostatic
potential in the presence of a membrane potential,
and coarse-grained dynamics
• Poisson-Nernst-Plank equations for Ion Permeation
through Membrane Channels
• Ion Permeation with coupled Dynamics of Membrane
Channels
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005
More Reading
Model Validation : Reaction Diffusion
• MCell Bartol and Stiles [2001]
• Continuum models Smart and McCammon
[1998]
•
Diffusion Simulations Naka et al
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
[1999]
September 2005
How do muscle cells function ?
Center for Computational Visualization
Institute of Computational and Engineering Sciences
Department of Computer Sciences
University of Texas at Austin
September 2005