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

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005

Molecules of the Cell

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005

Bacterial Cell

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

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!

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

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005

How do Nerve Cells Function ?

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005

Axonal transport of membranous organelles

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

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

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005

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

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

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

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

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005

Action Potential I

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005

Action Potential II

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.

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

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.

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!

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 ?

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

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]

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

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

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)

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

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

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]

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

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.

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

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)

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

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

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

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

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

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005

Synaptic Cleft Modeling

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]

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

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005

Fas2 meets AChE

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.

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005

AChE Tetramer Conformations

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

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

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

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

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]

Center for Computational VisualizationInstitute of Computational and Engineering SciencesDepartment of Computer Sciences University of Texas at Austin September 2005

How do muscle cells function ?