september 2005 center for computational visualization institute of computational and engineering...
TRANSCRIPT
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Lecture 7: Multiscale Bio-Modeling and Visualization
Cell Structures: Membrane and Intra-Cellular Molecule Models (NMJ)
Chandrajit Bajaj
http://www.cs.utexas.edu/~bajaj
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Molecules of the Cell
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Bacterial Cell
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Neuromuscular Junction (NMJ)
http://fig.cox.miami.edu/~cmallery/150/neuro/neuromuscular-sml.jpg
Movie!
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
How do Nerve Cells Function ?
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Axonal transport of membranous organelles
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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 ofvoltage-gated cation channels
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Action Potential I
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Action Potential II
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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.
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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 thepostsynaptic membrane toward the threshold potential for firing an action potential.
Inhibitory neurotransmitters open either Cl- channels or K+ channels, and this suppresses firing bymaking it harder for excitatory influences to depolarize the postsynaptic membrane.
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Neuromuscular Junction (NMJ)
http://fig.cox.miami.edu/~cmallery/150/neuro/neuromuscular-sml.jpg
Movie!
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
How do Synapses Occur at the Neuro-Muscular Junction ?
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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]
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
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
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Extracting Domain Information from Imaging data
• Cellular Membrane Geometry can be extracted (meta-balls)
• 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
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Synaptic Cleft Geometry
Twin resolution models for the Ce
From 14813 vertices and 29622 triangles to 4825 vertices and 9636 triangles
(~67% decimation)
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Acetylcholinesterase in Synaptic Cleft
• Activity Sites
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Activity Sites
AchE molecule (PDBID = 1C2B)
Cell MembraneEnlarged View
Datasets from www.pdb.org and Dr. Bakers group
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Nictonic-Acetyl-Choline Receptor
Pentameric Symmetry in AchR molecule (PDBID: 2BG9)Image from Unwin [8]
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
AChBP (1I9B.pdb) Active Sites
Complementary component
Primary Component
ACh Binding Site
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Specificity
• Ion channels are highly specific filters, allowing only desired ions through the cell membrane.
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Populating the Membrane with the molecules
Name PDBID Size (oA) Weight(kDa)
Density(/µm2)
Number-Atoms
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 - 50000
18
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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
Thin-plate spline interpolation with centers at local max & min
1139 centers, 9.55% error (middle); 7649 centers, 7.36% error (right)
Original MapRBF Approximation (5891 centers, 7.88%
error)
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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
• p = polynomial of degree k-1
• = Radial basis function (thin-plate spline kernel)
• Make Coefficients orthogonal to polynomials of degree k-1
),( ixx
),()()(1
i
M
ii xxxpxs
0)(1
M
iii xq
i
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
• One choice for
It minimizes “bending energy”:
It is conditional positive definite• Memory storage
• Computational cost
),( 21 xx
Thin-Plate Spline Kernel
221
2
||||
log)(
xxr
rrr
)( 2NO
)( 3NO
2
222
R
yyxyxx dxdyffffI
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
),(),(),(
),(),(),(
),(),(),(
21
22212
12111
MMMM
M
M
xxxxxx
xxxxxx
xxxxxx
A
Matrix Form
~~
00A
cP
PAsT
finitepositivedeA
finitepositivedenonA
~~
~
i
jiij
i
xpP
s
)(
function value at xi
, where pi(x) forms a basis for polynomial of degree k-1
coefficients of the RBF kernel at xi
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Matrix A (1065x1065)
Condition number = 2.95E+06 (non-positive definite)
Poor Conditioning
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
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)
Use of Pre-conditioners/Sparsifiers
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Synaptic Cleft Modeling
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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]
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Modeling Physiology I :Electrostatics Potential
dielectric properties of the solute and solvent, ionic strength of the solution ,
à r á["(rr V(r)]+ k2(r)sinh(V(r)) = ú(r)
k2
ú(r)
"(r)
solute atomic partial charges.Poisson-Boltzmann
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Fas2 meets AChE
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Adaptive Boundary Interior-Exterior Meshes
(a) monomer mAChE (b) cavity (c) interior mesh
(d) exterior meshes•Y. Zhang, C. Bajaj, B. Sohn, Special issue of Computer Methods in
Applied Mechanics and Engineering (CMAME) on
Unstructured Mesh Generation, 2004.
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
AChE Tetramer Conformations
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Model Physiology II
Reaction Diffusion Models• Time dependent equations to model the diffusion of ACh across the synaptic cleft
Initial Condition
Boundary Conditions
On the domain
at the AchR boundaries
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
Steady State Smulochowski Equation
(Diffusion of multiple particles in a potential field)
bbulk rprp
for )(
• -- entire domain
• -- biomolecular domain
• -- free space in
• a – reactive region
• r – reflective region
• b – boundary for
BC)(Robin )()()(or
for BC)(Dirichlet 0)(
rprrpJn
rrp a
rxrpJn for 0)(
Diffusion-influenced biomolecular reaction rate constant :
bulkp
dSrpJnk a
)(
)]()()()[()( rUrprprDrpJ
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.00E+000
5.00E+011
1.00E+012
1.50E+012
2.00E+012
2.50E+012
3.00E+012
3.50E+012
4.00E+012
4.50E+012
R
ate
(M-1m
in-1)
I (M)
1C2O 1C2B Int2 Monomer*2 Monomer*3 Monomer*4
Active Sites of AChE
•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
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment 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 [1999]
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Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005
How do muscle cells function ?