Statistical Physics of Self-Replication
Review of paper by Jeremy England
Mobolaji Williams — Shakhnovich Journal Club — Nov. 18, 2016
1
Theoretical Physics of Living Systems
Ambitious: Develop new mathematical/
principle based model of biological
process. (e.g., Michaelis Menten, HodgkinHuxely)
Conservative: Use our understanding of
physics to place limits on biological
processes.
“Can diffusion place limits on how fast
bacteria should swim?”
Edward Purcell, “Life at Low Reynolds
Number”,1976
Yes! They should swim faster than 30
microns/sec to outrun diffusion.
2
Self-Replication is Irreversible and Thus Produces Entropy
Self Replication
“One unit turns into two
units over some period of
time”
England’s Argument
— Self replication is irreversible
— Irreversible processes are associated
with increases in entropy
➤
Physics of entropy production (i.e., statistical physics)
should place thermodynamic bounds on self replication.
3
Irreversibility and Entropy through Master Equation - Part 1
— Transitions to and from
a single state
⇡(i1 ! j; t)
(j, t)
(i1 , t)
⇡(j ! i1 ; t)
— Transitions to and from
all possible states
..
.
(i1 , t)
(j, t)
(i2 , t)
..
.
(in , t)
..
.
4
Irreversibility and Entropy through Master Equation - Part 2
— Transitions to and from
all possible states
..
.
(i1 , t)
(j, t)
(i2 , t)
..
.
(in , t)
..
.
Xh
@
p(j, t) =
⇡(i ! j; t)p(i, t)
@t
i
[“Master Equation”]
⇡(j ! i; t)p(j, t)
i
— Foundational equation of Non-Equilibrium Statistical Physics
5
Irreversibility and Entropy through Master Equation - Part 3
For long times the system goes into equilibrium and is time independent
h
lim ⇡(i ! j; t)p(i, t)
t!1
i
⇡(j ! i; t)p(j, t) = 0
By thermal physics and by definition
e Ei
lim p(i, t) =
t!1
Z( )
lim ⇡(i ! j, t) ⌘ ⇡1 (i ! j)
t!1
[“Boltzmann Distributed Energies”]
[“Time Independent Transition Amplitude”]
Thus we have
⇡1 (j ! i)
=e
⇡1 (i ! j)
Ei!j
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Irreversibility and Entropy through Master Equation - Part 3
By conservation of Energy
bath
Qbath +
Qbath
Esystem =
Euniverse = 0
,
and so we have
system
⇡1 (j ! i)
=e
⇡1 (i ! j)
Esystem
i!j
Sbath
.
The definition of (dimensionless) entropy
in terms of heat:
S=
⇡1 (j ! i)
=e
⇡1 (i ! j)
Qi!j
➤
Q/kB T
The more irreversible the reaction,
the more entropy produced in the
forward direction
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Generalizations and Definitions
Let i, j, k,.. define a microstate, and I,
II, III, … define the macrostate.
Self-Replication Process
(A single unit of I)
Macrostate of I, can
be in any microstate i.
➤

⇡(II ! I)
ln
⇡(I ! II)
(Two units of I)
⇡(I ! II)
⇡(II ! I)
Macrostate of II, can be in
any microstate j.
[Defines the irreversibility of a self replication process]
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Second Law of Thermodynamics — Part 1
Some probability theory and algebra leads to…
Entropy Change of System
Heat Released into Bath

⇡(II ! I)
h Qibath, I!II + ln
+
⇡(I ! II)
Entropy Change Associated with ReactionIrreversibility
Conceptually
“Second Law of thermodynamics with macroscopic
irreversibility correction”
Ssys, I!II
bath
0
h Qibath, I!II
system
Ssys, I!II
⇡(I ! II)
⇡(II ! I)
➤
This equation applies in a wide range of far from
equilibrium scenarios!
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Formal to Informal Dictionary
Say we have a replicator which grows
at a rate g and degrades at a rate
n(t)
n(t) = n0 e(g
)t
t
We can then define
⇡(I ! II) = g dt
⇡(II ! I) = dt
[Probability of one unit to
duplicate in time dt]
[Probability of one unit to
degrade in time dt]
sint
[Entropy change over time dt]
q
[Heat released over time dt]
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Second Law of Thermodynamics — Part 2
q+
q+
Heat Released
sint
sint
ln
hg i
ln
hg i
[Main Equation of Paper]
Growth Rate
Degradation Rate
Internal Entropy Change
Two ways to look at this result:
1) Maximum Ratio: Given the amount of heat produced in a replication, what is
the minimum ratio between growth and decay rate?
2) Minimum Heat Released: Given a growth rate and a decay rate, what is the
minimum amount of heat released during replication/growth?
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Maximum Net-Growth Rate (i.e., Fitness)
We can also compute
Net Growth Rate
Degradation Rate
gmax
Maximum Fitness
= (exp[
q+
Heat Released
sint ]
1)
Internal Entropy Change
“Replicator’s maximum potential fitness is set by
how it exploits energy in its environment to
catalyze it’s own reproduction.”
Reproductive Fitness
depends on
Efficient Metabolism
12
Application: Bacterial Cell Division
Main Qualitative Prediction of Work
gmax
Low Durability of Replicator
= (exp[
1/
Large Heat Produced
q
Large Entropy Change
sint
}
q+
sint ]
Efficient Metabolism
1)
Reproductive
Fitness
13
Application: Self-Replicating Polynucleotides
Self Replicating RNA Enzyme:
+ substrates
Doubling Time: 1 hour
ln
Lincoln, Tracey A., and Gerald F. Joyce, Science (2009)
Cleavage Rate (i.e. Half Life): 4 years
Thompson, James E., et al. Bioorganic chemistry(1995)
hg i

4 years
= ln
1 hr
* Change in Internal Entropy: Negligible (Mixing
Entropy of Reactants ~ Mixing Entropy of Products)
➤
h Qi
RT ln(g/ ) = 7 kcal/mol
Experimentally, the heat
released is 10 kcal/mol
Minetti, Conceição ASA, et al. "Proceedings of the
National Academy of Sciences(2003)
Replicating RNA enzyme
operates close to
thermodynamic efficiency
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Application: Self-Replicating Polynucleotides
(Hypothetical!) Self Replicating Single Stranded DNA:
+ substrates
Doubling Time: 1 hour
Cleavage Rate (i.e. Half Life): 3 X 10^7 years
ln
hg i
4 years
= ln
1 hr
Schroeder, Gottfried K., et al. PNAS of the United States of America 103.11(2006)
➤
h Qi

RT ln(g/ ) = 16 kcal/mol > enthalpy of ligation reaction
DNA self-replication (of this kind)
is forbidden thermodynamically!
Does this relate to why RNA
(and not DNA) was the selfreplicating nucleic acid on
prebiotic earth?
15