Come on – Feel the Noiseor

Viscoelastic Force Spectra of Single Biomolecules:Mapping the Energy Landscape of Conformational

Transitions

Bhavin Khatri and

Tom McLeish

Polymer & Complex Fluids GroupSchool of Physics & Astronomy

A physicist’s definition of a biopolymer?

• Biopolymer monomers can adopt different conformations

• Different conformations have different energies and sizes– e.g. Dextran: Chair Boat: extra length, but cost in energy

• Fluctuations on energy landscape Viscoelasticity of transitions??

Protein ConcatamersPolysaccharides

Chair Boattransition

Probing Biopolymers:Conventional Force Spectroscopy

Features in force extension trace indicate conformational transitions

Something interesting

FJC

Spectroscopy of Proteins

• Relaxation time of monomers slower than experimental time=> pulling at different speeds can give global dynamical information; e.g. rate of unfolding of protein

• Cannot easily access local dynamical information; e.g. rubbing of helices and sheets

200

150

100

50

0

Forc

e [

pN

]

160140120100806040200

Extension [nm]

[Data:M. Kawakami]

Recall Polymer Rheology

Non-linear experiments came before ….

.. linear ones

Force Clamp Thermal Noise Spectroscopy

Photo Detector

LASER Diode

Polysaccharide chain

Cantilever

Piezoelectric Stage

1.0

0.8

0.6

0.4

0.2

x10-3

5040302010

PID

SetPoint

A

Power Spectral Density

Frequency [kHz]

8x10-4

6

4

2

403530252015105

PSD

[nm

2 /kHz

]Free Cantilever

320pN

620pN

920pN

Need model for power spectra…

Example:real PSD of cellulose

Data: M. Kawakami

Rouse with Internal Friction: RIF Model

Like Rouse but mode friction:

=> Internal friction important for short chains:

D. McInnes (1977) Polymer,18,505de Gennes, Scaling Concepts

Brownian Response of a 2-state monomer

• Brownian response due to

• Populations obey

• Linear response solution

Frictional Freely Jointed Chain

• Joints with constant friction and at high stretch

Torque

Restoring Force

From statistical mechanics

Fits to Internal Friction Spectrum

• Friction is underdetermined for dextran– Minima in elasticity & friction

– Geometry of landscape– Hence:

100

80

60

40

20

0

16001400120010008006004002000

Force [pN]

Detailed Balance is obeyed

Dextran

Cellulose

Reconstruction of Energy Landscape

• Brownian Fluctuations give inherent viscoelastic information– Example: Overdamped Spring & Dashpot

Dissipation and Dynamics

time

Force

Input: Brownian ‘kicks’

Power spectrum: frequency distribution of fluctuations

Elasticitydominates

Frictiondominates

Output: Sum over responsestime

Displacement

Idea• Probe local dissipation and dynamics• Noise in AFM experiments usually detrimental

– watch single molecule fluctuations under controlled force

• Fingerprint of dynamics on energy landscape• Analogy: macroscopic rheology of complex fluids

Overview of rest of talk

• Force Clamp Thermal Noise Spectroscopy• Coarse-grained biopolymer models• Molecular scale models• Comparison to experiment• Reconstruction of dextran energy

landscape

Spectroscopy of Polysaccharides

• Polymers with ringed monomers (e.g. glucose)

2500

2000

1500

1000

500

0

220210200190180170

Force

Extension

CelluloseDextran

3000

2500

2000

1500

1000

500

0

240220200180160

Force

Extension

No hystereris observed => relaxation time of chain faster than experimental timeCannot probe dynamics of monomers; only eqm elastic info [Data:M. Kawakami]

Rouse model

• Simplest model Spring and dashpot– But biomolecules are actually polymers

• Spectrum of relaxation times

Solvent Friction

Diffusion Equation

100

101

102

103

104

10-3

10-2

10-1

100

101

102

103

104

10-3

10-2

10-1

100

101

102

103

104

10-3

10-2

10-1

100

101

102

103

104

10-3

10-2

10-1

100

101

102

103

104

10-3

10-2

10-1

Frequency Response of End to End Vector

• For i >> R spring & dashpot (-1)

• For R >> i Rouse behaviour (-1/2)

– up to c ~1/i where internal friction of high modes dominate

Rouse

Spring & Dashpot

AFM experimentsEnd-End Vector Important

PSD of Cantilever and RIF polymer

• Cantilever response

• Combined parallel response

• Fluctuation Dissipation Theorem

Folding Funnel (Onuchic, Wolynes,..)

R

RN SOr Hyperspace?

Fitting to RIF Model to Experimental PSD

8x10-4

6

4

2

PS

D [n

m2 /k

Hz]

403530252015105Frequency [kHz]

Extract from experiments:

[Data:M. Kawakami]

Results:Viscoelastic Force Spectra

Solid lines are gradient of Force-Extension experiment

These biopolymers are ‘short’

End-End Solvent Friction [gkHz]

Force [pN]

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

140012001000800600400200

Force [pN]

Monomer Internal Friction [gkHz]200

150

100

50

0

1400120010008006004002000

Dextran

Cellulose

Force [pN]

Monomer Elasticity [pN/nm]70000

60000

50000

40000

30000

20000

10000

0

1400120010008006004002000

Cellulose

Dextran

Expect

[Data:M. Kawakami & K. Byrne]

Viscoelasticity on a 2-state landscape

• Identify Elasticity and Friction in terms of microscopic parameters

Depends only on eqm populations and obeys equipartition

Depends only on hopping time Like an asymmetric 1D lattice diffusion process

• Frequency response of monomer length ~ spring and dashpot

Overview of rest of talk

• Force Clamp Thermal Noise Spectroscopy• Coarse-grained biopolymer models• Molecular scale models• Comparison to experiment• Reconstruction of dextran energy

landscape

Fluctuations on a 2-state landscape

• Chain of 2-state monomers (1 dimensional)– Average length Equilibrium populations:

– Fluctuations of length hopping:

0 200 400 600 800 1000 1200 1400 1600 1800 200010

3

104

105

106

107

108

109

1010

1011

Physical Interpretation: Elasticity spectrum

• Hopping elasticity entropic in origin applying force changes effective size of ‘box’• Can measure zero force ΔG and Δx

0 200 400 600 800 1000 1200 1400 1600 1800 200010

0

101

102

103

104

105

106

Physical Interpretation:Internal Friction Spectra

• Force controls barriers heights and thus friction• In principle can measure:

Experiments again..• Microscopic model explains minima in elasticity

and friction

• Can explain elasticity spectrum qualitatively• Low force friction?• Bond friction?

Force [pN]

Monomer Elasticity [pN/nm]70000

60000

50000

40000

30000

20000

10000

0

1400120010008006004002000

Force [pN]

Monomer Internal Friction [gkHz]200

150

100

50

0

1400120010008006004002000

??

Modelling Entire Force Regime

• FJC

• Conformational hopping

• Bond stretching

• All independently additive to total extension

Fit to Elasticity Spectrum

• Very good agreement between experiment & model – Agree with literature values

• Minima in elasticity: entropically favourable to elongate

Dextran

Cellulose

35000

30000

25000

20000

15000

10000

5000

0

16001400120010008006004002000

Force [pN]

Reconstruction of Landscape

Dynamics controlled by barrier -> Internal Friction Force Spectrum

?

Implications..

• Bond & joint friction 6-7 orders of magnitude larger than expected solvent friction

– One explanation:Diffusion in rough potential (R.Zwanzig (1988), PNAS, 85, 2029)

• Barrier curvature– Discrete Kramer’s prefactor

Very sharp!

Protein experiments

• Future: controlled viscoelastic force spectra of refolding proteins

0.4

0.3

0.2

0.1

0.0Con

cata

mer

Inte

rnal

Fric

tion

[gk

Hz]

90807060504030Force [pN]

A

B10

8

6

4

2

Con

cata

mer

Ela

stic

ity [p

N/n

m]

90807060504030Force [pN]

126124122120118116E

xten

sion

[nm

]

302520151050Time [secs]

200

150

100

50

0

Forc

e [

pN

]

160140120100806040200Extension [nm]

12

3

[Data: M. Kawakami]

Summary• Brownian Noise can give detailed viscoelastic information of

single molecules• RIF model: a generic coarse-grained model of for

biopolymer with internal transitions• 2-state model provides insight to viscoelasticity of

conformational transitions• Viscoelastic force spectra reveal the statics and dynamics

on the conformational energy landscape of biomolecules– Reveals a dominance of internal friction in the nanoworld

• Future experiments on proteins may probe dynamics of secondary and tertiary structure formation in protein folding

• Thanks to Masaru Kawakami, Katherine Byrne and Alastair Smith for doing the AFM experiments!

• Thanks to EPSRC for funding.

References, Experiment: Langmuir, 401,400 (2004) ; Theory: submitted to Nature Physics

Modelling Entire Force Regime

• Each process spring & dashpot in nature, and independently additive to total extension

• Hence, at low frequency

Mechanical Oscillation Experiments

Humphris, Tamayo & MilesLangmuir, Vol. 16, No. 21, 2000

Dextran

Relaxation in RIF Model

100

101

102

10-2

10-1

100

101

102

103

Rouse

RIF