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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
1
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
12
x
y
t = 0
t = ⌧
x
⌧0
Detailed Balance
13
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
15
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
17
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
21
x
y
v
d
f
� ⌘ v/d ⌘ ⌘ ��xy
/��xy
⌘ f/A
Fix Predict Conclude
22
F (X) = �kBT ln
Z
x2Xdx e��E(x)
peq(X) / e��F (X)
Variational Principle for Macroscopic Steady States
23
F (X) = �kBT ln
Z
x2Xdx e��E(x)
peq(X) / e��F (X)
X
p eq(X
)
Variational Principle for Macroscopic Steady States
24
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⇤
25
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
26
t0
W (X)
W(=
V��xy
)
Excess Work in Driven Steady States
27
t0
W (X)
W(=
V��xy
)
Excess Work in Driven Steady States
28
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
32
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
35
p(W
|!X)
Wex
(X1
) Wex
(X2
)
�ex
=�2
2
⇥hW 2ic!X � hW 2ic
ss
⇤
for Gaussian distribution
Work Fluctuations
36
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
39
�⇤xy
= ���⌧V h�2xy
ieqpss(X) / e��[F (X)�Wex
(X)]
�
�⇤xy
Thermodynamic Prediction of Shear Stress
40
pss(X) / e��[F (X)�Wex
(X)]
�
�⇤xy
⌧� =⌧0
1 + k�⌧0
�⇤xy
= ���⌧�
V h�2xy
ieq
Thermodynamic Prediction of Shear Stress
41
pss(X) / e��[F (X)�Wex
(X)]
�
�⇤xy
⌧� =⌧0
1 + k�⌧0
�⇤xy
= ���⌧�
V h�2xy
ieq
Thermodynamic Prediction of Shear Stress
42
pss(X) / e��[F (X)�Wex
(X)]
�
�⇤xy
log �
log⌘
⌧� =⌧0
1 + k�⌧0
�⇤xy
= ���⌧�
V h�2xy
ieq
Open Questions
43
• 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
44
Thermal Equilibrium and Detailed Balance
Driven Steady States and Extended Linear Response
Driven Steady States in Biological Materials
45
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
46
Two Questions
• What triggers initiation of uncoating process?
• What is energetic cost of combining mechanical strength with rapid response?
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48
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
52
Wmin = �F
53
Speed-Strength TradeoffSp
eed
k on
e���F
Strength �F
�E
W
�F
Free
Ene
rgy
Driven steady state opens new possibilities
54
Wor
k
55
Chaperone couples assembly to chemical energy source
ATP
56
Chaperone couples assembly to chemical energy source
ATP
ADP
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Chaperone couples assembly to chemical energy source
Q
ATP
ADP
58
Chaperone couples assembly to chemical energy source
Q �µ = kBT lncATP
cADP+Q
59
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
61
Coarse-grain to one dimension
m = 1m =2
3m =
7
9m =
8
9
w+(m)
w�(m)
j(m)
m
f(m
)
62
Solve analytically in large N limit
pss(m) ⌘ N e��Nf(m)
m
f(m
)
63
Solve analytically in large N limit
pss(m) ⌘ N e��Nf(m)
m⇤
m
f(m
)
64
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
65
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
67
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
⇤ )
68
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 =
69
Open Questions
70
• 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