Elastic rod models for natural and synthetic polymers: analytical solutions for arc-length dependent...
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Transcript of Elastic rod models for natural and synthetic polymers: analytical solutions for arc-length dependent...
Elastic rod models for natural and synthetic polymers: analytical Elastic rod models for natural and synthetic polymers: analytical solutions for arc-length dependent elasticitysolutions for arc-length dependent elasticity
Silvana De Lillo, Gaia Lupo, Matteo SommacalSilvana De Lillo, Gaia Lupo, Matteo SommacalDipartimento di Matematica e Informatica and CR-INSTM VILLAGE Università di Perugia Dipartimento di Matematica e Informatica and CR-INSTM VILLAGE Università di Perugia
INFN sezione di PerugiaINFN sezione di Perugia
MMario Argeri,Vincenzo Baroneario Argeri,Vincenzo BaroneDipartimento di Chimica and CR-INSTM VILLAGE, Università Federico II di NapoliDipartimento di Chimica and CR-INSTM VILLAGE, Università Federico II di Napoli
CNR-IPCF PisaCNR-IPCF Pisa
11
22
Properties of different helices
Helix radius
Pitch Res.x turn
Rise x res.
A - DNAB - DNAZ - DNA
1.3 nm1.0 nm0.9 nm
2.46 nm3.32 nm4.56 nm
10.710.412.0
0.23 nm0.33 nm0.38 nm
-helix3/10 helix-helixCollagen
0.23 nm0.19 nm0.28 nm0.16 nm
0.54 nm0.60 nm0.47 nm0.96 nm
3.63.04.33.3
0.15 nm0.20 nm0.11 nm0.29 nm
33
A, B, (right-handed helices) and Z (left-handed helix) forms
of DNA
44
PRION DISEASESPRION DISEASESPRION DISEASESPRION DISEASES
Key eventKey event: : CONFORMATIONAL TRANSITIONCONFORMATIONAL TRANSITION
ClassificationClassification: neurodegenerative diseases: neurodegenerative diseases
Pathogen: Pathogen: PrPPrPScSc
PrPPrPCC
Which is the mechanism underlying conformational transition Which is the mechanism underlying conformational transition
of PrPof PrPCC to PrP to PrPScSc??Which is the mechanism underlying conformational transition Which is the mechanism underlying conformational transition
of PrPof PrPCC to PrP to PrPScSc??
Which factors do enhance the conformational transitionWhich factors do enhance the conformational transition??Which factors do enhance the conformational transitionWhich factors do enhance the conformational transition??
??PrPPrPScSc amyloid-like fibrilsamyloid-like fibrils
--helix 40 %helix 40 % 30 %30 %--sheet 3 %sheet 3 % 43 %43 %
55
Representative conformations of infinite homopolypeptidesRepresentative conformations of infinite homopolypeptides
a
b
c
-sheet(C5)-sheet(C5)
227 7 ribbonribbon
331010 helix (C10) helix (C10)
helixhelix
66
Teflon [a homopolymer (CFTeflon [a homopolymer (CF22))nn ] ]forms 13/6 and 15/7 helicesforms 13/6 and 15/7 helices
c=16.97 c=16.97
77
The numerical model: atomistic The numerical model: atomistic simulationssimulationsGeneral Liquid Optimized Boundary (GLOB) General Liquid Optimized Boundary (GLOB)
G.Brancato, N.Rega, V.Barone, J.Chem.Phys.G.Brancato, N.Rega, V.Barone, J.Chem.Phys. 124, 214505 124, 214505 (2006).(2006).
The external wallThe external wall Constant volumeConstant volume
Bulk reaction fieldBulk reaction field
The Buffer regionThe Buffer region
Bulk densityBulk density
No preferential orientationNo preferential orientation
88
bulkbulk
Nucleosome – DNA complexabout 3x102 DNA bases; 6x104 water molecules: 2x105 atoms
99
1010
The analytical model: elastic stripThe analytical model: elastic stripA.Goriely, M.Tabor, Phys.Rev.Lett. 77,3537-3540 A.Goriely, M.Tabor, Phys.Rev.Lett. 77,3537-3540 (1996)(1996)
In most cases the environment of the helix axis is anisotropic. 1111
The arc length is given by
tdtd
dz
td
dy
td
dxdsts
tt
0
222
0
For an helix we get
tcRtdcR
tdctRtRts
t
t
22
0
22
0
222 cossin
1212
ds
Tdsk
F
)(
The Frenet curvature kkFFss measures the shift from a rectilinear behaviour: it is defined as the modulus of the derivative of the tangent vector w.r.t. the arc length
CurvatureCurvature
2
222
2
p
R
R
cR
R
ds
Tdsk
F
The curvature of a circular The curvature of a circular helix is CONSTANThelix is CONSTANT
1414
The Frenet torsion Fs measures the shift from a planar behaviour
For a circular helix
2
222
2
2
p
R
p
cR
cs
F
The torsion of a circular helixThe torsion of a circular helixis CONSTANTis CONSTANT
1515
x
y
z
O
The strip is characterized bya non null transverse sectionand is subjected to suitabledeformations
Select possible deformations and dynamic variablesSelect possible deformations and dynamic variables Select the forces coming into playSelect the forces coming into play Write the equations associated to static equilibrium configurationsWrite the equations associated to static equilibrium configurations
and determine the geometric shapeof these configurationsand determine the geometric shapeof these configurations 1616
Deformations Deformations (not allowed in our (not allowed in our model)model)
x
y
z
O
Compression,Compression,lengtheninglengthening
shearshear
UndeformedUndeformedconfigurationconfiguration
1717
Deformations Deformations (allowed in our (allowed in our model)model)
x
y
z
O
2 orthogonal bendings2 orthogonal bendings Torsion (twist)Torsion (twist)
UndeformedUndeformedconfigurationconfiguration
1818
KinematicsKinematics
x y
z
O
3d
1d
2d
sr
The elastic strip isdescribed by:
3: ERIr
passing through the centersof the transverse sections
A generalized Frenet frame
sdsdsd 321 ,,
A curve
21,dd Define the plane of the
Transverse section
1919
Dynamic Dynamic variablesvariables
x y
z
O
1d
2d
sr
3d
321 ,, ddd
The frameis orthonormal, so thata vector (Darboux sk
vector) exists that describes the variation of sd
i
3,2,1 isdsksdii
3
1i
iisdksk
21,kk describe the bendingbending
3k describes the twisttwist 2020
x
y
z
O
td
3
2d 1d
b
n
td
bnd
bnd
3
1
1
cossin
sincos
ds
dk
kk
kk
F
F
F
3
2
1
cos
sin
The two frames are The two frames are related by a rotation ofrelated by a rotation ofAngle Angle around around
Describes theintrinsic twistintrinsic twist
sr
3d
2121
ds
dk
kkk
F
F
3
22
21
ForcesForces
x y
z
O
sr
A resulting force A resulting momentum
sF
sM
Internal Efforts Internal Efforts equivalent to
Possibly external forcesexternal forces(gravity, friction) equivalent to
Resulting external force Resulting external momentum
sf
s
sfsF
'
ssFsrsM
'
sr
In general the action of these forcesIn general the action of these forcesdetermines a movement describeddetermines a movement describedby non banal equationsby non banal equations
On the transverse section placed in act:
2424
Eqilibrium equationsEqilibrium equations
x y
z
O
0
0
Fds
rd
ds
Mdds
Fd
sr
In the absence of external forces at equilibrium we get
0
0
3
Fdds
Mdds
Fd
2525
33222111 dbkdkadkasM
Bending stifnessBending stifness Twist Twist stifnessstifness
Jb
EIa
EIa
22
11
22
21
22
21
1
1
4
11
2
22
A
AE
A
A
IJ
IA
J
Rod (with radius A)Rod (with radius A)
Elliptic strip (withElliptic strip (with
semiaxes Asemiaxes A11,A,A22))
E = Young modulus; E = Young modulus; = Shear modulus; = Shear modulus;
II11,I,I22 = principal inertia moments in the cross-section plane = principal inertia moments in the cross-section plane 2626
Equilibrium equations: Equilibrium equations: constitutive relationships constitutive relationships
DKds
Dd
Fdds
Mdds
Fd
ˆˆˆ
0
0
3
0
0
0ˆ
),,(ˆ
12
13
23
321
332211
kk
kk
kk
K
dddD
dFdFdFF
T
2727
212133
13112
22
2
23221
11
1
12213
31132
23321
)(
)(
)(
kkaads
dbk
ds
dkb
Fkkbads
dak
ds
dka
Fkkbads
dak
ds
dka
kFkFds
dF
kFkFds
dF
kFkFds
dF
2828
The Lancret’s theorem
22
22
2
p
R
R
cR
Rk
F2
222
2
2
p
R
p
cR
cF
A helix is a curve, whose tangent makes a constant angle with a fixed lineIn terms of the Frenet frame defined by the so called tangent, normal, andbinormal vectors:
(1)
)()()( sBsNsF
ds
sTdsk
F
)()(
For a general helix For a general helix Lancret’s theorem Lancret’s theorem
states thatstates that
For a circular For a circular helixhelix
),,( BNT
)(
)(
s
sk
F
F
2929
A circular helixcircular helix is describedby the parametric equation
p
ct
tR
tR
t sin
cos
r
R
cp 2
-2-1
0 12
-2-1
012
0
5
10
15
-2-1
012
t
ct
tR
tR
tr sin
cos
R RadiusRadius of the circular cylinder of the circular cylinderalong which the curve is coiledalong which the curve is coiled
c
R
““Speed” of Speed” of advancementadvancementalong the helix axis. along the helix axis. PitchPitch of the helix, i.e. of the helix, i.e. distance between two distance between two successive spires. successive spires.
cp 23030
A. Goriely, M. Nizette, M. Tabor, J. Nonlinear Sci. 11,3-45 (2001)
The (“inverse problem”) approach:- Most of the helices we are interested in are circular
helices (kF and F constant);
- We assign constant values to kF and F ;- We choose the function ;- We solve Kirchhoff’s equations for the six unknowns
F1 , F2 , F3 , a1 , a2 , b
with fixed “initial” values ;
• a1 , a2 , b constant
a1 = a2 (circular rod) leads to arbitrary
a1 ≠ a2 (generic rod) leads to n
• We obtain many new results, in both cases
= and ≡ (s) .
• We recover all the results already present
in literature with a1 , a2 , b constant.
• Energy landscape (variational principle)
• Time evolution
• Two-dimensional limit (ribbon)
Work in progressWork in progress