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10/3/2003
Molecular and Cellular Modeling
10/3/2003
Introduction
• Objective: to construct a comprehensive simulation
software system for the computational modeling of
mechanical interactions in proteins and cells.
• Main idea: civil engineering approach to biology
• Mathematical model: thin rod theory – Ribbons and groups
– Perturbation method
– Finite element model
10/3/2003
Ribbons and Groups
• Ribbons are framed paths: (r,n) central curve director
• Deformation: The set of deformation forms Lie group under
• Rotation matrix and rotation rate Rotation matrix Rotation rate
3R L] [0, :r 3R L] [0, :n
M), , (a, d ,R a 3 (3) L] [0, :M(s) SO,R D
)MM, ,a(a)M , ,)(aM , ,(a 212121222111 ,)(r)M(ear(0) d(r)(s)
0 s
d M(s)n(s)d(n)(s)
I MMT
MM v, v :,, TT321
10/3/2003
Perturbation Method
• Equilibrium configurations: minimum energy state
Quadratic approximation• Thin rod equations
• Perturbation equations
L
0 el
L
0 exelex )),(),(d((J))),d(r((J ),(E (d(r))E E(d) dssssdsss
s
dttt0 00 )()uM(a(0)r r(s)
Mu r (1) 0 M M (2)
r
J (3) ex
MQ-u)(M )'(Q (4) T0
T
Qcs T
2
1),,(J 2
el
TM M whereM and vM )Q( )'(Q T tt
QM-Mu)( v T0 tTT
10/3/2003
Finite Element Model
• Strains and strain increments
• Energy increments
1i1i1i-1ii p)xp(Ry
1iTii MM of raterotation the
Tiiii M)M( and x :increments
T1i1iii1ii,
i
i ] x x[Dy
I0
0I
I-0
VI-
M0
0MD 1ii,
Ti
Ti
1ii,
i
iii
i
iiii
yKy
yKy
2
1E
ii
10/3/2003
Molecular Mechanics - Proteinmorphosis
• Objective: to create a general model for ligand binding
and more general protein conformational changes.
• Microscopic approaches: molecular dynamics
• Main features of our system– macroscopic formulation– protein backbone as 1-D elasticity thin rod– global parameterization– incremental energy minimization– interactive visualization
10/3/2003
Protein Structure Hierarchy
• Protein: linear amino acid polymer polypeptide chains
• Primary: amino acid sequence
• Secondary: helix, sheet
• Supersecondary: motif
• Tertiary: single polypeptide
• Quaternary: i.e. enzyme complex
10/3/2003
Hierarchical Representation Primary structure Secondary structure
Tertiary structure Quaternary structure
10/3/2003
Dynamic Visualization
10/3/2003
Collision Vector Visualization
10/3/2003
Global Parameterization• Backbone - rather constrained degrees of freedoms:
– Fixed bond length, bond angle
– Torsion angles are freedoms– Every third bond is rigid
• Sidechains - less flexibility.
• Local coordinate: torsion angles
• Global coordinate: displacement of nodal atoms.
C
C
C
N
N
H
H
H
C’
C’
O
O
sidechain
10/3/2003
Potential Energy
• Force field: torsion, hydrogen bond, van der Waals, distance restraint
• Stiffness and load
10/3/2003
Variational Approach
• Total stiffness and load
• Linear constraint quadratic minimization
• Multi-scale methods
c
Lx
B
BK T
0
10/3/2003
Calmodulin Peptide BindingCalmodulin Peptide binding (PDB)
Peptide binding (simulation)Animation
10/3/2003
Hemoglobin Oxygen Binding
• Hemoglobin structure:
symmetrically arranged four
subunits• Allosteric oxygen binding
Deoxygenated Hb (T state) Oxygenated Hb (R state)
10/3/2003
Modeling of Allostery
• Trigger of allosteric transition
• Loss of cooperativity
F helixF8 HIS
Feheme
10/3/2003
Cellular Mechanics - Cellmate• From gel volume to civil engineering structure: cytoskeleton network and its three components
• How mechanics controls biochemistry
10/3/2003
Tensegrity - Architecture of Life • Continuous tension and isolated compression.• Prestressed tensions• Features:
- Highly stable
- Global response to a local signal
10/3/2003
Cells as Tensegrity Structures• Tension elements: microfilaments• Compression elements: microtubules and ECM• Intermediate filaments as integrators and guy wires
10/3/2003
Image Processing
• Data acquisition: confocal microscopy
• Filament extraction
10/3/2003
Create Virtual Cytoskeleton
10/3/2003
Reach-In Interface
10/3/2003
Model of Cell Crawling
10/3/2003
An Integrated System
• A multi-scale computational model
• Different virtual environment: CAVE, Reach-In
10/3/2003
Conclusions
• It is a multi-disciplinary research:
- mathematics and physics
- biology
- computer science
- engineering
• The result can also be used elsewhere