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Molecular and Cellular Modeling

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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

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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

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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

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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

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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

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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

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Hierarchical Representation Primary structure Secondary structure

Tertiary structure Quaternary structure

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Dynamic Visualization

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Collision Vector Visualization

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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

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Potential Energy

• Force field: torsion, hydrogen bond, van der Waals, distance restraint

• Stiffness and load

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Variational Approach

• Total stiffness and load

• Linear constraint quadratic minimization

• Multi-scale methods

c

Lx

B

BK T

0

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Calmodulin Peptide BindingCalmodulin Peptide binding (PDB)

Peptide binding (simulation)Animation

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Hemoglobin Oxygen Binding

• Hemoglobin structure:

symmetrically arranged four

subunits• Allosteric oxygen binding

Deoxygenated Hb (T state) Oxygenated Hb (R state)

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Modeling of Allostery

• Trigger of allosteric transition

• Loss of cooperativity

F helixF8 HIS

Feheme

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Cellular Mechanics - Cellmate• From gel volume to civil engineering structure: cytoskeleton network and its three components

• How mechanics controls biochemistry

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Tensegrity - Architecture of Life • Continuous tension and isolated compression.• Prestressed tensions• Features:

- Highly stable

- Global response to a local signal

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Cells as Tensegrity Structures• Tension elements: microfilaments• Compression elements: microtubules and ECM• Intermediate filaments as integrators and guy wires

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Image Processing

• Data acquisition: confocal microscopy

• Filament extraction

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Create Virtual Cytoskeleton

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Reach-In Interface

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Model of Cell Crawling

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An Integrated System

• A multi-scale computational model

• Different virtual environment: CAVE, Reach-In

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Conclusions

• It is a multi-disciplinary research:

- mathematics and physics

- biology

- computer science

- engineering

• The result can also be used elsewhere