Thermodynamics of single molecule stretching...Thermodynamics of single molecule stretching Author Eivind Bering, Signe Kjelstrup, Dick Bedeaux, Miguel Rubi, Astrid de Wijn Created

Thermodynamics of single molecule stretchingEivind Bering1, Signe Kjelstrup2, Dick Bedeaux2, Miguel Rubi3, Astrid de

Wijn4

1PoreLab/Department of Physics, NTNU2PoreLab/Department of Chemistry, NTNU

3Departament de Física Fonamental, Universitat de Barcelona4Department of Mechanical and Industrial Engineering, NTNU

August 30, 2019

Mo va on

• Do thermodynamics apply to small systems?• Many oppose that a e.g. Gibbs-Duhem rela onship is applicable for

single molecular systems1

• May have a large fundamental impact on how one can describebiomolecular and other events related to energy conversion on thesmall scale.

• The aim of the project is to verify the rate laws presented by Rubi et al.2

1David Keller, David Swigon, and Carlos Bustamante. “Rela ng Single-MoleculeMeasurements to Thermodynamics”. In: Biophysical Journal 84.February (2003),pp. 733–738.

2J M Rubi, D Bedeaux, and S Kjelstrup. “Thermodynamics for Single-Molecule StretchingExperiments”. In: Journal of computa onal chemistry 110.25 (2006), pp. 12733–12737.

2

Outline

• Mo va on• Background• Model and method• Previous work• Free energy es ma on• Rate laws and entropy produc on• Conclusion

3

The rate laws

There is a fric on coefficient associated with an increment of eitherstretching velocity 𝑣 or the constraining force 𝑓 .

In the Helmholtz ensemble (length controlled)

Δ 𝑓 = 𝜉𝑙(𝑙)𝑣, (1)

and in the Gibbs ensemble (force controlled)

Δ𝑓 = 𝜉𝑓(𝑓) 𝑣. (2)

4

Hill’s theory for small systemthermodynamics3

• An equilibrium descrip on of thermodynamic proper es on thenano-scale.

• The thermodynamic func ons cease to be extensive at this length scale.• Hill introduced an ensemble of small systems, and used the replica

energy to obtain thermodynamic proper es that depended on thesurface area and curvatures.

• The choice of ensemble will affect the thermodynamic poten als forsmall systems.

3Terrell L. Hill. Thermodynamics of Small Systems. New York: W.A. Benjamin Inc, 1963,p. 210.

5

Polyethylene oxide (PEO)

• It is studied extensively experimentally.• Applica ons ranging from industrial manufacturing to medicine.• A be er understanding of the mechanical proper es of PEO on the

nano-scales is needed.

Polyethylene Oxide (PEO) force field

United atom (UA) force field developed by van Zon et al. (2001)based on a explicit atom force field by Neyertz et al.(1994)

Intermediate scattering function calculated from this UA model is ingood agreement with neutron spin-echo experiments

A. van Zon, B. Mos, P. Verkerk, and S. W. de Leeuw, Electrochim. Acta2001, 46 (10), 1717 doihttps://doi.org/10.1016/S0013-4686(00)00776-3 S. Neyertz, D.Brown, and J. O. Thomas, J. Chem. Phys 1994, 101, 10064, doihttps://doi.org/10.1063/1.467995

E. Bering (NTNU) MD of polymeric fibre bundles February 15, 2018 16 / 46

6

Previous work: Breaking of bundles

10 nm

7


10 nm

8


10 nm

9


10 nm

10


10 nm

11


10 nm

12


10 nm

13

Simula on details

• United atom force field• Stretching, bending, torsion• Langevin thermostat at T=200 K

14

Two ensembles

Helmholtz ensemble (isometric)

𝑦𝑧

𝑙

Gibbs ensemble (isotensional)

𝑓(−ez) 𝑓(ez)

15

Model valida on using the entropicregime

The entropy in the molecule is given by

𝑆(𝑙) − 𝑆0 = −32𝑘B𝑙2/𝑅2

g (3)

giving rise to an entropic force 𝑓S of

𝑓S = −𝜕(𝑆(𝑙)𝑇 )𝜕𝑙 = 3𝑘B𝑇 𝑙/𝑅2

g . (4)

In the entropic region the molecule follows a random walk in threedimensions. If we let the molecule be composed of 𝑁eff independent unitsof length 𝑏eff, the radius of gyra on reads

𝑅g = 𝑙ext√6𝑁eff, (5)

where 𝑙ext = 𝑁eff𝑏eff is length of the fully extended molecule.16


This gives an effec ve entropic force of

𝑓S = 18𝑘B𝑇 𝑙𝑁eff

𝑙2ext(6)

The number of independent units depends on the system size, whereas theeffec ve length 𝑏eff should be size independent.

17


−0.010

0.010.020.030.040.050.06

0 5 10 15 20 25 30 35 40

𝑓(nN

),𝑓(

nN)

𝑙 (Å), 𝑙 (Å)

IsometricIsotensional

18


−0.010

0.010.020.030.040.050.06

0 5 10 15 20 25 30 35 40

𝑓(nN

),𝑓(

nN)

𝑙 (Å), 𝑙 (Å)

IsometricIsotensional𝑓𝑆 with 𝑁𝑒𝑓𝑓 = 9.0 and 𝑏𝑒𝑓𝑓 = 8.9

19

Es ma on of free energyFor the isometric simula ons, there is a fluctua ng force for each length. Ifwe let

⟨𝑓(𝑡)⟩𝑙 = 𝑓(𝑙), (7)

the Helmholtz energy is given by

𝐴(𝑙) = ∫𝑙

𝑙0

𝑓(𝑙′)d𝑙′. (8)

For the isotensional simula ons there is a fluctua ng length for each force. Ifwe let

⟨𝑙(𝑡)⟩𝑓 = 𝑙(𝑓), (9)

the Gibbs free energy is given by

𝐺(𝑓) = ∫𝑓

𝑓0

𝑙(𝑓′)d𝑓′. (10)

20

Es ma on of free energy

In the thermodynamical limit 𝐴 and 𝐺 are related by a Legendretransforma on, and with Δ𝑙 = 𝑙 − 𝑙0 and Δ 𝑓 = 𝑓(𝑙) − 𝑓(𝑙0) one has

𝐴(𝑙) + 𝐺(𝑓 = 𝑓(𝑙)) = Δ 𝑓Δ𝑙 (11)

for sufficiently large systems4.


21


In the thermodynamical limit 𝐴 and 𝐺 are related by a Legendretransforma on, and with Δ𝑙 = 𝑙 − 𝑙0 and Δ 𝑓 = 𝑓(𝑙) − 𝑓(𝑙0) one has

𝐴(𝑙) + 𝐺(𝑓 = 𝑓(𝑙)) = Δ 𝑓Δ𝑙 (11)

for sufficiently large systems4.

Let us inves gate how this is for a PEO chain of 𝑁 = 51 units.


21

Force-elonga on

0

1

2

3

4

5

0 10 20 30 40 50 60 70

𝑓,𝑓(

nN)

𝑙, 𝑙 (Å)22

Force-elonga on

0

1

2

3

4

5

0 10 20 30 40 50 60 70

0

0.5

1

50 55

𝑓,𝑓(

nN)

𝑙, 𝑙 (Å)


23


05

10152025303540

0 10 20 30 40 50 60 70

𝐴(𝑙),

Δ𝑓Δ

𝑙−𝐺(

𝑓 𝑡(

𝑙))(1

0−19

J)

𝑙, 𝑙 (Å)


24


05

10152025303540

0 10 20 30 40 50 60 70

2

4

6

8

50 55 60

𝐴(𝑙),

Δ𝑓Δ

𝑙−𝐺(

𝑓 𝑡(

𝑙))(1

0−19

J)

𝑙, 𝑙 (Å)


2

4

6

8

50 55 60

25

Entropy produc on

When changing the controlled force from 𝑓 to 𝑓𝑒𝑥𝑡, the entropy produc onhave previously been shown to be5

d𝑆d𝑡 = 1

𝑇 (𝑓𝑒𝑥𝑡 − 𝑓)d 𝑙d𝑡 . (12)

With Δ𝑓 = 𝑓𝑒𝑥𝑡 − 𝑓 , the resul ng force controlled rate law is

Δ𝑓 = 𝜉𝑓(𝑓)d 𝑙d𝑡 ≡ 𝜉𝑓(𝑓) 𝑣, (13)

where 𝜉𝑓=𝜉𝑓(𝑓) is the isotensional fric on coefficient and 𝑣 the resul ngaverage stretching velocity.


26

Entropy

The entropy produc on associated with increasing the length of themolecule at a constant velocity 𝑣 = d𝑙

d𝑡 is given by

d𝑆d𝑡 = 1

𝑇 ( 𝑓𝑒𝑥𝑡 − 𝑓)d𝑙d𝑡 ≡ 𝜉𝑙(𝑙)𝑣, (14)

and with Δ 𝑓 = 𝑓𝑒𝑥𝑡 − 𝑓 as the velocity induced force change, the lengthcontrolled rate law is

Δ 𝑓 = 𝜉𝑙(𝑙)d𝑙d𝑡 ≡ 𝜉𝑙(𝑙)𝑣, (15)

27

Force controlled rate law

Our first aim is to determine the force controlled rate law:

Δ𝑓 = 𝜉𝑓(𝑓) 𝑣. (16)

The samples are equlilibrated for 5 ns at a fixed force before the force isincreased by a step of Δ𝑓

28

Force controlled rate law

Our first aim is to determine the force controlled rate law:

Δ𝑓 = 𝜉𝑓(𝑓) 𝑣. (16)

The samples are equlilibrated for 5 ns at a fixed force before the force isincreased by a step of Δ𝑓

Let us look at a the average length as a func on of me for samplesequlilibrated at 𝑓0 = 0.67 nN when we increase the force by 8% at 𝑡 = 0.

28

Force controlled simula ons:Δ𝑓 = 𝜉𝑓(𝑓) 𝑣

56.2

56.3

56.4

56.5

56.6

56.7

−10 −5 0 5 10

𝑙(Å)

𝑡 (ps)29


56.2

56.3

56.4

56.5

56.6

56.7

−10 −5 0 5 10

𝑙(Å)

𝑡 (ps)

𝑙𝑓𝑖𝑡 = 0.35𝑡 + 𝑙0(here Δ𝑓 = 0.05 nN)

30


0

0.05

0.1

0.15

0.2

0.25

0.3

0 20 40 60 80 100

Δ𝑓(

nN)

𝑣 (m/s)

𝑓0 = 1.00 nN𝑓0 = 0.67 nN𝑓0 = 0.33 nN

31


0

0.05

0.1

0.15

0.2

0.25

0.3

0 20 40 60 80 100

Δ𝑓(

nN)

𝑣 (m/s)

𝑓0 = 1.00 nN𝑓0 = 0.67 nN𝑓0 = 0.33 nN

Δ𝑓 = 0.0016 𝑣Δ𝑓 = 0.0013 𝑣Δ𝑓 = 0.0009 𝑣

32

Lenght controlled rate law

Our next aim is to determine the length controlled rate law:

Δ 𝑓 = 𝜉𝑙(𝑙)𝑣. (17)

The samples are equlilibrated for 5 ns at a fixed length before the length isincreased at a constant velocity.

33

Lenght controlled rate law

Our next aim is to determine the length controlled rate law:

Δ 𝑓 = 𝜉𝑙(𝑙)𝑣. (17)

The samples are equlilibrated for 5 ns at a fixed length before the length isincreased at a constant velocity.

Let us look at a the average force as a func on of me for samplesequlilibrated at 𝑙0 = 52 Å when we stretch the molecule at a constantvelocity 𝑣 = 4 m/s from 𝑡 = 0.

33

length controlled simula ons:Δ 𝑓 = 𝜉𝑙(𝑙)𝑣

0

0.1

0.2

0.3

0.4

0.5

−40 −30 −20 −10 0 10 20 30 40

𝑓(nN

)

𝑡 (ps)

𝑓 = 0.21

34


0

0.1

0.2

0.3

0.4

0.5

−40 −30 −20 −10 0 10 20 30 40

𝑓(nN

)

𝑡 (ps)

𝑓 = 0.21𝑓fit(𝑡) = 0.0035𝑡 + 0.22

35


−0.06−0.05−0.04−0.03−0.02−0.01

00.010.020.030.040.05

2 4 6 8 10 12 14

Δ 𝑓(nN

)

𝑣 (m/s)

Figure: The es mated ∆ 𝑓 associated with stretching molecules of length 𝑙0 = 52Å for stretching veloci es 2 − 14 m/s, shown with a fi ng curve from linearregression. All data points are averaged over 200 samples.

36


−0.06−0.05−0.04−0.03−0.02−0.01

00.010.020.030.040.05

2 4 6 8 10 12 14

Δ 𝑓(nN

)

𝑣 (m/s)

𝑓fit(𝑣) = 0.0014𝑡 − 0.013

Figure: The es mated ∆ 𝑓 associated with stretching molecules of length 𝑙0 = 52Å for stretching veloci es 2 − 14 m/s, shown with a fi ng curve from linearregression. All data points are averaged over 200 samples.

37

Fric on coefficents

−1−0.5

00.5

11.5

22.5

3

20 25 30 35 40 45 50 55 60 65 70

𝜉(10

−12

kg/s

)

𝑙, 𝑙 (Å)

Length controlledForce controlled

38

Entropy produc on at 𝑣 = 14 m/s

−1−0.5

00.5

11.5

22.5

3

20 25 30 35 40 45 50 55 60 65 70

𝜉(10

−12

kg/s

)

𝑙, 𝑙 (Å)

Length controlledForce controlled

39

Conclusion

• Nanometric chains of PEO have been analyzed in the Helmholtz andGibbs ensemble by non-equilibrium molecular dynamic simula ons.

• On this length scale, the thermodynamic poten als are ensembledependent

• They s ll tell a coherent story in agreement with the rate lawspresented by Rubi et al.6

• A thermodynamic descrip on is meaningful also on the small scale.


40

Conclusion






40

Conclusion






40

Conclusion






40

Thank you!

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Documents

Thermodynamics of single molecule stretching...Thermodynamics of single molecule stretching Author Eivind Bering, Signe Kjelstrup, Dick Bedeaux, Miguel Rubi, Astrid de Wijn Created