Modeling of Biofilaments: Elasticity and Fluctuations Combined D. Kessler, Y. Kats, S. Rappaport (Bar-Ilan) S. Panyukov (Lebedev) Mathematics of Materials and Macromolecules IMA, Minneapolis, October 3, 2004

Stretching of helical springs Overview 1. Motivation 2. Ribbons: geometry, elasticity, fluctuations 3. Computer simulations: Frenet algorithm Stretching of filaments Twisting dsDNA Cyclization Distribution functions

Polymers – objects with atomic thickness (1 A) and arbitrary length Atomic resolution Quantum mechanics RIS models Coarse grained description Statistical mechanics Random walks

What sort of objects are described by this model? This is the probability distribution of a random walk! Beads connected by entropic springs The standard model of polymers: n n-1 spring constant= kT/l 2

Random walks are not lines! 2 s R(s)R(s) 0 L Continuous curve: Inextensible line Random walk: Extensible fractal

What about nano-filaments: thickness A? 1Intrinsic shape 2Resistance to change of shape (bending, twist) Biofilaments: DNA, actin and tubulin fibers, flagella, viruses … Synthetic filaments: organic microtubules, carbon nanotubes, … But : thermal fluctuations are still important! Theory of elasticity of fluctuating filaments with arbitrary intrinsic shape New elements:

2 Bending elasticity of inextensible lines Modeling dsDNA at large deformations Bustamante et al., Science 265, 1599 (1994) The first step:

dsDNA under stretching and torque 1.Cannot twist lines 2.Lines have no chirality degree of over/unwinding Strick et al., Science 271, 1835 (1996)

Geometry of space curves: s s’ t t n n b b Frenet eqs : generate curve by rotation of the triad - curvature,- torsion This is not a physical twist !

Helix p 2r Straight line Circle r

Ribbons (stripes) t 2 (s) t 1 (s) Physical triad: t 1, t 2, t 3 n(s) b(s)

Generalized Frenet eqs. – rotation of physical axes Ribbon - principal axes ; tangent 5 Configuration of the ribbon – uniquely defined by or by rate of twist

Mechanics: Linear Elasticity Deviations from stress-free state : Elastic Energy - rigidity with respect to bending and twist Small local but arbitrarily large global deviations from equilibrium configuration! 6 Equilibrium shape defined by spontaneous curvatures

Stretching a helical spring pitch > radius, bending rigidity > twist rigidity 4 turns, minimize

pitch < radius, bending rigidity < twist rigidity Phys. Rev. Lett. 90, (2003) The energy landscape E(R) has multiple minima with depths and locations that vary with F

Stretching helical ribbons of cholesterol: Smith, Zastavker and Benedek, Phys. Rev. Lett. 87, (2001) Mechanical noise-induced transitions?

Stretching transitions and hysteresis in chromatin ? Y. Cui and C. Bustamente, PNAS 97, 127 (2000).

Correlation functions for ribbons with arbitrary spontaneous shape and rigidity! 7 - random Gaussian variables Fluctuation energy: Thermal Fluctuations - persistence lengths Phys. Rev. Lett. 85, 2404 (2000) Phys. Rev. E 62, 7135 (2000)

Weak fluctuations of a helix: e1e1 t 3 (s ) t 2 (s ) t 1 (s ) s e3e3 ( ) e2e2 Persistence lengths > helical period frequency

Ribbon with spontaneous twist – model for dsDNA? 13 Europhys. Lett. 57, 512 (2002)

Buckling under torsion: stability diagram

Frenet-Based Computer Simulations 1. Generate random numbers 2. Integrate Frenet eqs. to generate configurations 3. Excluded volume, attractive interactions – Boltzmann weights Direct simulation of fluctuating lines! Phys. Rev. E (2002)

Rectilinear rod L=2

Does twist affect conformation? is independent of twist ! Exact result: if there is no spontaneous curvature - WLC model ok ?

Rectilinear ribbon Twist affects conformation! J. Chem. Phys. 118, 897 (2003) L=2

What about objects with spontaneous curvature? Consider small deformations of a planar ring y x 2r

Twist and bending fluctuations – always decouple, but: for curved filaments – twist is not simply rotation of cross-section! Example: small fluctuations of a planar ring andTwist rigidity -coupling between(rotation)(conformation) zero-energy modes Out-of-plane fluctuations diverge! (vanishes for )

Euler Angles s/r

Open Ring 1 Probability T= Fluctuation-induced shape transitions – at fixed local curvature! elastic moduli

Length L=1.5 Effect of spontaneous curvature on cyclization Probability of R End-to-end distance R cyclization

Fundamental Exponent

Effect of constant spontaneous curvature

Effect of random spontaneous crvature

Effect of twist rigidity on cyclization of curved filaments

Stretching fluctuating filaments Unbiased sampling of configurations – works only for small f f How are fluctuations affected by the force?

Large-scale fluctuations are suppressed by stretching

MS approximation breaks down for short filaments with L<a (neglect orientational effects)! L=6.28

All orientations are equally probable Flexible chain Rigid filament No Wall

f=1f=10f=2f=3 y x End fluctuations of stretched filaments: simulation results Experiments: short dsDNA segments (ca 1000 bp) actin filaments

Take home message: Bending rigidity is not enough! New generation of models of biofilaments that account for : intrinsic shape (spontaneous curvature and twist) twist rigidity