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The Edge of Thermodynamics: Driven Steady States in

Physics and Biology

Robert Marsland England Lab, MIT

IGERT Summer Institute May 31, 2017

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Physics of Living Systems

The general struggle for existence of animate beings is not a struggle for raw materials – these, for organisms, are air, water and soil, all abundantly available – nor for energy which exists in plenty in any body in the form of heat, but a struggle for [negative] entropy, which becomes available through the transition of energy from the hot sun to the cold earth.

— Ludwig Boltzmann, 1875

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3

4

5

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Thermal Equilibrium and Detailed Balance

Driven Steady States and Extended Linear Response

Driven Steady States in Biological Materials

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Thermal Equilibrium

8

Thermal Equilibrium

9

Thermal Equilibrium

p(A)

p(B)

10

Thermal Equilibrium

Tx

peq(x) =1

Ze�

E(x)kBT

Detailed Balance

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Detailed Balance

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x

y

t = 0

t = ⌧

x

⌧0

Detailed Balance

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x

y

P[x⌧0 ] = peq(x0)p[x

⌧0 |x0]

t = 0

t = ⌧

x

⌧0

P[x⌧0 ] = peq(x⌧ )p[x

⌧0 |x⌧ ]

Detailed Balance

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y

x

P[x⌧0 ] = P[x⌧

0 ]

Detailed Balance

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y

x

p[x⌧0 |x⌧ ]

p[x⌧0 |x0]

=peq(x0)

peq(x⌧ )= e

�EkBT

x

T (1), {µ(1)i }

T (2), {µ(2)i }

T(3

) ,{µ

(3)

i}

T(4

) ,{µ

(4)

i}

Local Detailed Balance

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p[x⌧0 |x⌧ ]

p[x⌧0 |x0]

= e�Q

kBTp[x⌧

0 |x⌧ ]

p[x⌧0 |x0]

= e��SekB

G. Crooks, 1999 J. Schnakenberg, 1976

Summary

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Probability calculations are easy in equilibrium

Nothing happens in equilibrium

Equilibrium probabilities constrain driven dynamics

p(A)

p(B)

x

18

Thermal Equilibrium and Detailed Balance

Driven Steady States and Extended Linear Response

Driven Steady States in Biological Materials

Driven Steady States of Colloidal Suspensions

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Centre for Industrial Rheology

Quantitative Description

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x

y

v

d

f

� ⌘ v/d ⌘ ⌘ ��xy

/��xy

⌘ f/A

Quantitative Description

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x

y

v

d

f

� ⌘ v/d ⌘ ⌘ ��xy

/��xy

⌘ f/A

Fix Predict Conclude

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F (X) = �kBT ln

Z

x2Xdx e��E(x)

peq(X) / e��F (X)

Variational Principle for Macroscopic Steady States

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F (X) = �kBT ln

Z

x2Xdx e��E(x)

peq(X) / e��F (X)

X

p eq(X

)

Variational Principle for Macroscopic Steady States

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F (X) = �kBT ln

Z

x2Xdx e��E(x)

peq(X) / e��F (X)

X

p eq(X

)

Variational Principle for Macroscopic Steady States

X⇤

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F (X) = �kBT ln

Z

x2Xdx e��E(x)

peq(X) / e��F (X)

limV!1

peq(X) = �(X �X⇤)

p eq(X

)

X

Variational Principle for Macroscopic Steady States

X⇤

Excess Work in Driven Steady States

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t0

W (X)

W(=

V��xy

)

Excess Work in Driven Steady States

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t0

W (X)

W(=

V��xy

)

Excess Work in Driven Steady States

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t0

W (X)

W(=

V��xy

)

Excess Work in Driven Steady States

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hW(t)i

!X

W (X)

t 0

Excess Work in Driven Steady States

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hW(t)i

!X

W (X)

hW iss

t

⌧

0

Excess Work in Driven Steady States

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hW(t)i

!X

W (X)

hW iss

t

⌧

0

Wex

(X)

Excess Work in Driven Steady States

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hW(t)i

!X

W (X)

hW iss

t

⌧

Wex

= ⌧ [W (X)� hW iss

]

0

Driven Steady-State Distribution

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F (X) = �kBT ln

Z

x2Xdx e��E(x)

peq(X) / e��F (X)

Driven Steady-State Distribution

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p[x⌧0 |x⌧ ]

p[x⌧0 |x0]

= e�Q

kBT

R. Marsland and J. England, 2015

F (X) = �kBT ln

Z

x2Xdx e��E(x)

peq(X) / e��F (X)

pss(X) / e��[F (X)�Wex

(X)]��ex

(X)

Work Fluctuations

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p(W

|!X)

Wex

(X1

) Wex

(X2

)

�ex

=�2

2

⇥hW 2ic!X � hW 2ic

ss

⇤

for Gaussian distribution

Work Fluctuations

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Wex

(X1

) Wex

(X2

)

�ex

constant pss(X) / e��[F (X)�Wex

(X)]

p(W

|!X)

Work Fluctuations

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�ex

constant

R. Marsland and J. England, 2015

Additive noise

X(t)

X = �1

⌧(X �Xss) + ⇠(t)

Thermodynamic Prediction of Shear Stress

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�⇤xy

= ���⌧V h�2xy

ieqpss(X) / e��[F (X)�Wex

(X)]

Thermodynamic Prediction of Shear Stress

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�⇤xy

= ���⌧V h�2xy

ieqpss(X) / e��[F (X)�Wex

(X)]

�

�⇤xy

Thermodynamic Prediction of Shear Stress

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pss(X) / e��[F (X)�Wex

(X)]

�

�⇤xy

⌧� =⌧0

1 + k�⌧0

�⇤xy

= ���⌧�

V h�2xy

ieq

Thermodynamic Prediction of Shear Stress

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pss(X) / e��[F (X)�Wex

(X)]

�

�⇤xy

⌧� =⌧0

1 + k�⌧0

�⇤xy

= ���⌧�

V h�2xy

ieq

Thermodynamic Prediction of Shear Stress

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pss(X) / e��[F (X)�Wex

(X)]

�

�⇤xy

log �

log⌘

⌧� =⌧0

1 + k�⌧0

�⇤xy

= ���⌧�

V h�2xy

ieq

Open Questions

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• Can nonequilibrium phase transitions be understood in terms of work rates and relaxation times?

• How should we analyze systems with high-affinity chemical reactions?

ATP ADP

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Thermal Equilibrium and Detailed Balance

Driven Steady States and Extended Linear Response

Driven Steady States in Biological Materials

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Download clathrin video at http://idi.harvard.edu/uploads/mm/images/

endocytosis_celldance_small.mov

Basics of Clathrin Dynamics

• Clathrin self-assembles into stable spherical lattice

• Clathrin lattice exerts force to bend membrane

• Hsc70 actively disassembles clathrin

ATP ADP

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Two Questions

• What triggers initiation of uncoating process?

• What is energetic cost of combining mechanical strength with rapid response?

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Uncoating triggered by membrane modification

PI(4,5)P2PI(4)PPI(3)PPI(3,4)P2

LipidsClathrin

K. He, R. Marsland et al. (under review)

time

conc

entra

tion

Two Questions

• What triggers initiation of uncoating process?

• What is energetic cost of combining mechanical strength with rapid response?

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Free

Ene

rgy

kon

e���F

�F

ckon

Binding energy controls ratio of rates

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Free

Ene

rgy

kon

e���F

�F

ckon

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Diffusion-limited on-rate independent of interaction

Binding energy determines mechanical resilience

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Wmin = �F

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Speed-Strength TradeoffSp

eed

k on

e���F

Strength �F

�E

W

�F

Free

Ene

rgy

Driven steady state opens new possibilities

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Wor

k

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Chaperone couples assembly to chemical energy source

ATP

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Chaperone couples assembly to chemical energy source

ATP

ADP

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Chaperone couples assembly to chemical energy source

Q

ATP

ADP

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Chaperone couples assembly to chemical energy source

Q �µ = kBT lncATP

cADP+Q

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Coarse-grain to one dimension

60

Coarse-grain to one dimension

c kon

, k ⌧ kon

e���F

c ko

n

k on

e���

F

k

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Coarse-grain to one dimension

m = 1m =2

3m =

7

9m =

8

9

w+(m)

w�(m)

j(m)

m

f(m

)

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Solve analytically in large N limit

pss(m) ⌘ N e��Nf(m)

m

f(m

)

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Solve analytically in large N limit

pss(m) ⌘ N e��Nf(m)

m⇤

m

f(m

)

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Solve analytically in large N limit

pss(m) ⌘ N e��Nf(m)

m⇤

0

0

4kBT 12kBT mf(m

)

0.02kon

0.03kon

NkBT

Compute cost of acceleration

Strength �F (m⇤)

Spee

dw

�(m

⇤ )

Dissipation rate j(m⇤)�µ

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0

0

4kBT 12kBT mf(m

)

0.02kon

0.03kon

NkBT

Compute cost of acceleration

Strength �F (m⇤)

Spee

dw

�(m

⇤ )

Dissipation rate j(m⇤)�µ

66

Compute cost of acceleration

Strength �F (m⇤)

Spee

dw

�(m

⇤ )

0

0

4kBT 12kBT mf(m

)

0.02kon

0.03kon

NkBT

Dissipation rate j(m⇤)�µ

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Compute cost of acceleration

Strength �F (m⇤)0

0

4kBT 12kBT mf(m

)

0.02kon

0.03kon

NkBT

Dissipation rate j(m⇤)�µ

Spee

dw

�(m

⇤ )

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Compute cost of acceleration

Strength �F (m⇤)

Spee

dw

�(m

⇤ )

0

0

4kBT 12kBT

0.02kon

0.03kon

NkBT

Dissipation rate j(m⇤)�µ

(k + kon

e���F )m⇤Max speed =

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Open Questions

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• What is the effect of finite dimensionality?

• What governs the emergence of new phases?

• Find relevant measures of strength in experimental systems.

m

f(m

)

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Free energy, work rate and relaxation time determine steady-state properties far from equilibrium when fluctuation dynamics are linear.

Dissipation of chemical energy accelerates response of strong self-assembled structures.

Conclusions

Thank you!

Have more questions? Want notification of preprints?

Email me at marsland@mit.edu

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