bhavin khatri and tom mcleish polymer & complex fluids group school of physics & astronomy
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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
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