Engineering polymer molecules: Twirling Engineering polymer molecules: Twirling DNA Rings and loop polymersDNA Rings and loop polymers

Rochish ThaokarIIT Bombay

Collaborators:Igor Kulic, University of PennsylvaniaHelmut Schiessel, Univ Leiden, The Netherlands

Nano-sized gadgets doing mechanical work

Nano gears, lever arms, wheels=> nanomachine

Can we make Nanomotors

NanomachinesNanomachines

MedicineMedicine

Cancer recognising nanovehicles

“repair” cells, bones,tissues and reinforcement

Kinesin motor proteins moving on

Microtubules

Are there biological nanomachinesAre there biological nanomachinesBiological molecular motors!!

Myosin and actin are responsible for

muscle contraction

What are the properties of the material What are the properties of the material making a nanomachinemaking a nanomachine

StabilitySelf assembly abilityModularity and ReplicabilitySwitchabilityExperimentally tractable

DNA

The DNA nanomachines available today!!The DNA nanomachines available today!!

DNA Hybridisation

Conformationalchanges

Four orders of magnitude slower than their biological counterparts

Large switching times: 103 s

Drawbacks of the current nanomachinesDrawbacks of the current nanomachines

Structural complexityCan we come up with a simpler nanomachineWith Sub-second switching timesSwimming as fast as a bacteria (100 µ/s)

A DNA MINIMPLASMID

Has three degrees of freedom

is the angle of rotation around its centerline

We are interested in the degree of freedom

In thermal equilibrium, the ring fluctuates in an unbiased manner

There is no directed motion

A DNA MINIMPLASMIDA DNA MINIMPLASMID

U(x)

Diffusion

x

x

U(x)

No Net Motion

x

U(x)

Basic Brownian Ratchet (Potential Basic Brownian Ratchet (Potential ratchet)ratchet)

No Net current for a symmetric potential

U(x)

Diffusion

x

x

U(x)

Net Motion

x

U(x)

Basic Brownian Ratchet (Potential Basic Brownian Ratchet (Potential ratchet)ratchet)

Energy input from the switching potential U(x) used to rectify random thermal motion

U(x)

Diffusionx

x

U(x)

Net Motion

x

U(x)

Basic Brownian Ratchet (Thermal ratchet)Basic Brownian Ratchet (Thermal ratchet)

Energy input from the switching Temperature T(t) used to rectify random thermal motion

T=Finite

T=0

T=0

Nature uses and designs the ratchet to extract useful work from Thermal fluctuations

Some biological molecular motors described by Brownian ratchets

The Kinesin-Microtubulemotion is ratchet motion: ATP provides the energy, +/-ends of microtubule ,conformation change, theasymmetry

Examples in NatureExamples in Nature

Can we make use of the ratchet principle Can we make use of the ratchet principle to move our DNA minplasmid ?to move our DNA minplasmid ?

The DNA is a complicated molecule:

Neucleotides, Deoxiribose sugar and Phosphate

Double Helix, Highly charged polymer

Coarse grain the DNA: Its a Semiflexible polymer

<t(s) t(s+l)> ~ 0 Flexible ~ Semiflexiblee l lp

What is the most general form of the What is the most general form of the hamiltonian??hamiltonian??

Consider a circular ring: = /2; =s/R

Consider a Thermal Ratchet:

T(t)=To (1+A Sin f t)

Current:

The Hamiltonian is asymmetric in variable

is a fluctuating variable and so governed by Fokker Planck equation

l1=l2 =45, 50 nm;

200 nm;

R/r=10;

R=10 nm

f= 1000 Hz

= 200 rad/s

The ring rotates around its centerlineThe ring rotates around its centerline

How can we heat up the DNA so fast?How can we heat up the DNA so fast?

Ultrasound

Metal Nanocrystals covalently attached to DNA

Can the twirling ring translate?Can the twirling ring translate?

Fluid Mechanics !! Typical Reynolds numbers: 10^(-3)

Low Reynolds number hydrodynamics!!

All the fundamental solutions satisfy stokes equation

The torus should be force free and torque free

Construct solution consisting rotlets:

Stokes Fundamental solutionStokes Fundamental solution

How fast does the ratchet moveHow fast does the ratchet move??50 nm/s

Potential ratchet: E=Eo(1+A sin f t)

A can be as high as 0.3

Can be realised by operating the system close to DNA duplex melting

temperature

The ring moves at around 5 microns/second!!The ring moves at around 5 microns/second!!

As good as the speed of an amoebaAs good as the speed of an amoeba

Generates Torques (kbT) and forces (fN)Generates Torques (kbT) and forces (fN)

note: note: Rotational diffusion can alter the courseRotational diffusion can alter the course

Life at Low Reynolds NumberLife at Low Reynolds Number

Inertial negligible, Quasi-static ,reversible equations ref : Life at Low Reynolds number : Purcell

Comparison of Torus translational velocity

DNA on rails

Interaction between two torri

Can there be flow induced organisation

ConclusionsConclusions

A DNA Miniplasmid can be a NanomachineA DNA Miniplasmid can be a Nanomachine

Intrinsic curvature and anisotropy for asymmetry in potentialIntrinsic curvature and anisotropy for asymmetry in potential

A Thermal ratchet can cause it move at around 50 nm/sA Thermal ratchet can cause it move at around 50 nm/s

A potential ratchet can make it move at 10 micros/sA potential ratchet can make it move at 10 micros/s

Switching times is Smoluchowski times and so very fastSwitching times is Smoluchowski times and so very fast

We can possibly explain Purcell's swimming animalWe can possibly explain Purcell's swimming animal

References

1.1. Igor M Kulic, Igor M Kulic, RochishThaokarRochishThaokar, Helmut Schiessel, , Helmut Schiessel, “Twirling DNA “Twirling DNA rings- Swimming Nanomotors ready for kick start”,rings- Swimming Nanomotors ready for kick start”, Europhysics Letters, Europhysics Letters, 72,527-533, 200572,527-533, 2005

2.2. I.M.Kulic, I.M.Kulic, Rochish ThaokarRochish Thaokar, Helmut Schiessel, , Helmut Schiessel, “A DNA ring acting as a thermal ratchet”, J Phys Cond Matt, 17, S3965, 2005 J Phys Cond Matt, 17, S3965, 2005

3.3. Rochish ThaokarRochish Thaokar, Igor Kulic, Helmut Schiessel, , Igor Kulic, Helmut Schiessel, ““Hydrodynamics of a rotating torus”, submitted to Physics of Fluids submitted to Physics of Fluids

Force extension study of Loop Force extension study of Loop DNADNA

Collaborators:

Herve Mohrbach, Univ Leiden, The Netherlands

Vladimir Lobaskin, MPIP-Mainz Germany

Help study the structure and conformation of individual molecules

Where do these loop polymers occur:Where do these loop polymers occur:

a) Freely sliding linker proteins stabilizes a DNA loop

b) A rigid ligand with opening angle alpha causes a kink in DNA

c) Tangentially anchored DNA stretched by an AFM tip

Why Study force-extension curveWhy Study force-extension curve

Manipulations by a magnetic trapManipulations by a magnetic trap

Small magnets allow for stretching and twisting a DNA molecule. By measuring the distance of the bead to the surface and its fluctuations x2, one deduces the DNA’s extension, l and the stretching force, F.

magnets

magnetic bead

DNA

surfaceF =

k BT

2 l

Theory of Force-extension curves Theory of Force-extension curves

Straight chainsStraight chains

T=0 state has Enthalpy Ho

Thermal Fluctuations lead to entropic contributions

H=Ho+H Q=Qo*Q1

Q1 is entropic part of partition functionQ1 is entropic part of partition function

How to calculate the partition function How to calculate the partition function for fluctuationsfor fluctuations

Polymer problems can be suitably mapped to quantum mechanical problems

A vast literature exists for QM problems

An analogy and variable transformation can directly give the Partition function

Quantum Mechanics

Polymer Physics

Variable

transformatio

n

Calculation of Q1Calculation of Q1

Exponential Partition function contributes to Force extension

Wormlike chain model: lp is the persistence length

Used to Fit DNA, Proteins etc.

Looped chainsLooped chains

sqrtkbT lp/F) The enthalpic part is non-trivial

Inplane-out of plane fluctuations contribution Inplane-out of plane fluctuations contribution

to entropic part to entropic part Is there a quantum mechanical analogueIs there a quantum mechanical analogue

The eigen values allow us to evaluate the partition functionThe eigen values allow us to evaluate the partition function

There is discrete and continuous spectrum of eigen valuesThere is discrete and continuous spectrum of eigen values

Q=Q(Enthalpic,loop)*Q(Entropic,loop)

Q(Entropic,kink) =Q(Entropic,straight)*Q1

Q=QQ=Q(Enthalpic,loop(Enthalpic,loop)*Q)*Q(Entropic,loop)(Entropic,loop)

QQ(Entropic,kink(Entropic,kink)) =Q =Q(Entropic,straight)(Entropic,straight) *Q1 *Q1

Q=QQ=Q(Enthalpic, loop(Enthalpic, loop))*Q*Q(Entropic,straight)(Entropic,straight) *Q1 *Q1

Results:Results:

Q1Q1==sqrtsqrt((4 lp L/4 lp L/^2^2) ) Q1 is linearQ1 is linear

Partition function for the unstable loop can be Partition function for the unstable loop can be

resolvedresolved

The entropic contribution are logrithmicThe entropic contribution are logrithmic

(courtsey Herve Mohrbach)(courtsey Herve Mohrbach)

The force extension can be given by a WLCThe force extension can be given by a WLC

The apparent persistence length: l*=lp (1+8 lp/L)^(-2)The apparent persistence length: l*=lp (1+8 lp/L)^(-2)

The entropic contribution of kinks and loops is logrithmic in the high stretch limit

Enthlapic contribution is important and is easily calculated

The force extension curves for DNA with adsorbed proteins and the AFM tip

uncertainty can be addressed

Simulation vs theorySimulation vs theory

(courtsey Vladimir Lobaskin)(courtsey Vladimir Lobaskin)

Force-extension curvesForce-extension curves

ConclusionsConclusions

A semiflexible chain which is looped or non-straight can be A semiflexible chain which is looped or non-straight can be

expressed by a WLC modelexpressed by a WLC model

The persistence length is re-normalizedThe persistence length is re-normalized

The renormalisation is due to enthalpic effectsThe renormalisation is due to enthalpic effects

The theory explains discrepancies in the force-extension The theory explains discrepancies in the force-extension

curves semiflexible polymers, looped and kinked DNAs and curves semiflexible polymers, looped and kinked DNAs and

DNA protein complexesDNA protein complexes

Future WorkFuture Work

A Lattice Boltzmann-Brownian dynamics method to compute the flow fields for non-slender torus

Analytical and simulation model for interaction of two torri

Interaction of an ensemble of torri

Effect of thermal fluctuations on the single torus

References

Igor Kulic, H Mohrbach, V Lobaskin, Rochish Thaokar, Helmut Igor Kulic, H Mohrbach, V Lobaskin, Rochish Thaokar, Helmut Schiessel, Schiessel, “Apparent persistence length renormalisation of a Bent DNA”, Physical Review E, 72, 041905-1-5,2005 Physical Review E, 72, 041905-1-5,2005

Igor Kulic, H Mohrbach, Rochish Thaokar, Helmut Schiessel, Igor Kulic, H Mohrbach, Rochish Thaokar, Helmut Schiessel, “Equation of state of a looped DNA”,“Equation of state of a looped DNA”, Physical Review E (Under Physical Review E (Under Review)Review)