Elastic rod models for natural and synthetic polymers: analytical solutions for arc-length dependent elasticity Silvana De Lillo, Gaia Lupo, Matteo Sommacal Dipartimento di Matematica e Informatica and CR-INSTM VILLAGE Università di Perugia INFN sezione di Perugia Mario Argeri,Vincenzo Barone Dipartimento di Chimica and CR-INSTM VILLAGE, Università Federico II di Napoli CNR-IPCF Pisa 1

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Properties of different helices Helix radiusPitchRes.x turnRise x res. A - DNA B - DNA Z - DNA 1.3 nm 1.0 nm 0.9 nm 2.46 nm 3.32 nm 4.56 nm nm 0.33 nm 0.38 nm  -helix 3/10 helix  -helix Collagen 0.23 nm 0.19 nm 0.28 nm 0.16 nm 0.54 nm 0.60 nm 0.47 nm 0.96 nm nm 0.20 nm 0.11 nm 0.29 nm 3

A, B, (right-handed helices) and Z (left-handed helix) forms of DNA 4

PRION DISEASES Key event: CONFORMATIONAL TRANSITION Classification: neurodegenerative diseases Pathogen: PrP Sc PrP C Which is the mechanism underlying conformational transition of PrP C to PrP Sc ? Which is the mechanism underlying conformational transition of PrP C to PrP Sc ? Which factors do enhance the conformational transition ? ? PrP Sc amyloid-like fibrils  - helix 40 %30 %  - sheet 3 %43 % 5

Representative conformations of infinite homopolypeptides  -sheet(C5) 2 7 ribbon 3 10 helix (C10)  helix 6

Teflon [a homopolymer (CF 2 ) n ] forms 13/6 and 15/7 helices 7

The numerical model: atomistic simulations General Liquid Optimized Boundary (GLOB) G.Brancato, N.Rega, V.Barone, J.Chem.Phys. 124, (2006). The external wall Constant volume Constant volume Bulk reaction field Bulk reaction field The Buffer region The Buffer region Bulk density Bulk density No preferential orientation No preferential orientation 8  bulk

Nucleosome – DNA complex about 3x10 2 DNA bases; 6x10 4 water molecules: 2x10 5 atoms 9

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The analytical model: elastic strip A.Goriely, M.Tabor, Phys.Rev.Lett. 77, (1996) In most cases the environment of the helix axis is anisotropic. 11

The arc length is given by For an helix we get 12

We can express t as a function of the arc length s  and reparametrize the curve in terms of s 13

k F  s  The Frenet curvature k F  s  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 Curvature The curvature of a circular helix is CONSTANT 14

The Frenet torsion  F  s  measures the shift from a planar behaviour For a circular helix The torsion of a circular helix is CONSTANT 15

The strip is characterized by a non null transverse section and is subjected to suitable deformations  Select possible deformations and dynamic variables  Select the forces coming into play  Write the equations associated to static equilibrium configurations and determine the geometric shapeof these configurations and determine the geometric shapeof these configurations 16

Deformations (not allowed in our model) Compression,lengthening shear Undeformedconfiguration 17

Deformations (allowed in our model) 2 orthogonal bendings Torsion (twist) Undeformedconfiguration 18

Kinematics The elastic strip is described by: passing through the centers of the transverse sections A generalized Frenet frame A curve Define the plane of the Transverse section 19

Dynamic variables The frame is orthonormal, so that a vector (Darboux vector) exists that describes the variation of bending describe the bending twist describes the twist 20

The two frames are related by a rotation of Angle  around Describes the intrinsic twist 21

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inextensible strips For inextensible strips not deformable with shear we get Observation In this case the dynamic variables reduce to just 23

Forces   A resulting force   A resulting momentum Internal Efforts Internal Efforts equivalent to external forces Possibly external forces (gravity, friction) equivalent to   Resulting external force   Resulting external momentum In general the action of these forces determines a movement described by non banal equations On the transverse section placed in act: 24

Eqilibrium equations In the absence of external forces at equilibrium we get 25

Bending stifness Twist stifness Rod (with radius A) Elliptic strip (with Elliptic strip (with semiaxes A 1,A 2 ) semiaxes A 1,A 2 ) E = Young modulus;  = Shear modulus; I 1,I 2 = principal inertia moments in the cross-section plane 26

Equilibrium equations: constitutive relationships 27

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The Lancret’s theorem A helix is a curve, whose tangent makes a constant angle with a fixed line In terms of the Frenet frame defined by the so called tangent, normal, and binormal vectors: (1) For a general helix Lancret’s theorem states that For a circular helix For a circular helix 29

circular helix A circular helix is described by the parametric equation Radius of the circular cylinder along which the curve is coiled “Speed” of advancement along the helix axis. Pitch of the helix, i.e. distance between two successive spires. 30

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 (k F and  F constant); - - We assign constant values to k F and  F ; - - We choose the function  ; - - We solve Kirchhoff’s equations for the six unknowns F 1, F 2, F 3, a 1, a 2, b with fixed “initial” values ; a 1, a 2, b constant a 1 = a 2 (circular rod) leads to arbitrary  a 1 ≠ a 2 (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 a 1, a 2, b constant.

Energy landscape (variational principle) Time evolution Two-dimensional limit (ribbon) Work in progress