Thermodynamics of Error

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Thermodynamics of Error and Error Correction in
Brownian Tape Copying
C.H. Bennett and M. Donkor IBM Research Yorktown
17 April 2008
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For any given hardware environment, e.g. CMOS,
DNA polymerase, there will be some tradeoff among
dissipation, error, and computation rate. More
complicated hardware might reduce the error,
and/or increase the amount of computation done
per unit energy dissipated. This tradeoff is
largely unexplored, except by chemical engineers.
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Practical Fractional Stills
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Most studies of DNA and RNA polymerases tend to
focus on mechanisms for recognition of the site
and strand of the DNA to be copied, mechanisms
for binding, initiation, and termination of
copying, etc. rather than the copying process
itself. (cf WEHI animation see separate mov
file) Here we study the thermodynamics of some
simple Brownian copying engines modeled on
polymerases, but meant to elucidate the
speed-error-dissipation tradeoff at a fundamental
level.
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The chaotic world of Brownian motion, illustrated
by a molecular dynamics movie of a synthetic
lipid bilayer (middle) in water (left and right)
dilauryl phosphatidyl ethanolamine in
water http//www.pc.chemie.tu-darmstadt.de/researc
h/molcad/movie.shtml
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RNA polymerase reaction viewed as a
one-dimensional driven random walk
activation free energy
driving force
or as thermal diffusion on a washboard potential
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Tilting the washboard the other way (e.g. by
increasing the PP concentration) makes the
driving force negative, resulting in reversible
erasure or un-copying of an already synthesized
strand of RNA.
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Higher activation barrier for error transition
But, because the RNA strand separates from the
DNA original after leaving the copying enzyme,
error transitions have the same driving force as
good transitions.
driving force
Copying errors therefore act as reversible
obstructions, difficult to insert, and equally
difficult to remove.
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Copying with errors may be viewed as a random
walk on a binary tree
Random walk parameterized by p forward step
probability, and s selection factor against
errors
Activation barrier difference d kT ln
(1-s)/s
Here C denotes a correct base (in a 2-letter
alphabet for simplicity) and E an error base.
Error transitions (those that add or remove
errors) are less frequent.
Forward driving force e kT ln p/(1-p)
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Steady State Solution errors are uniformly and
randomly distributed in the copy tape with some
frequency h. Error rate h and the drift
velocity v are calculated as a function of random
walk parameters p and s. Solve
self-consistently for drift velocity v and
rates of error incorporation and removal, so the
net rate of incorporation equals the drift
velocity times the steady state error frequency
h. v h ps (1p)s h
net error incorporation rate equation
v p (1p) ((1s)(1h) sh )
drift velocity equation
make
unmake error
step forward
---------- step back---------
v and h as functions of p with s0.1

• For any sgt0,
• Steady state has
• v 0 and
• 1/2
• when p 1/3

drift velocity v
1 / 2
error rate h
0.1
forward step probability p
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v0
Uncopying regime vlt0. Uncopying rate depends
on tapes initial error concentration, being
slower the more errors originally present.
Copying regime vgt0. After a transient period
of copying or uncopying, copying rate becomes
independent of initial tape contents
Paradoxical regime where vgt0 despite negative
driving force.
Copying rate v
Forward step probability p
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Typical random walk trajectories showing how
errors impede uncopying
With errors s0.1 Step-right probability
p1/3 Leftward drift gets stalled by errors (red)
Without any possibility of errors s0 Forward
step probability p1/3 Trajectory drifts leftward
Time
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When intrinsic error rate s is low but nonzero,
the random walk trajectory distribution is
skewed, with a long tail on the left.
Time
For p between 1/3 and 1/2, the drift is initially
negative (uncopying), but changes to positive
after enough errors accumulate.
1000 walks of 4000 steps each s 0.01 p 0.40
First 720 steps
All 4000 steps, shrunk vertically
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Skewness and initial leftward drift disappear
if walks are initialized with an appropriate
concentration of errors. Left comet is for walks
with no initial errors. Right comet has
initial error concentration 0.34, from steady
state model s0.01, p0.4 N1000, L4000 as
before
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• Dissipation (entropy production per step) is a
  sum of 2 terms
• External entropy due to work done by the
  external driving force
• Dext v ln (p/(1-p)) / v
• Dext can be negative if v and ln(p/(1-p)) have
  opposite sign.
• Internal Shannon entropy of incorporated errors
  (has same sign as v))
• Dint v/v (-h ln(h) -(1-h)ln(1-h))
• Dissipation per step Dext Dint is
  nonnegative.

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dissipation per step s
1
copying speed v
dissipation per step s
10-2
incorporated error rate h
10-4
uncopying speed -v
h0 10-4
copying speed v
10-6
-ln 2
1
2
-1
-2
0
driving force e ln(p/(1-p))
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Proofreading in DNA Replication
Polymerase activity (1) tries to insert correct
base, but occasionally (2) makes an error.
Exonuclease activity (3) tries to remove errors,
but occasion-ally (4) removes correct bases. When
both reactions are driven hard forward the error
rate is the product of their individual error
rates.
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Discrimination s
f
(slower and/or more dissipative)
Dissipation mainly in external driving reactions
Dissipation mainly in incorporated errors. At
high error rate, this pushes process forward
even against uphill external driving force
h
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Non-proofreading kinetics is degenerate,
reflecting locality of error processes in
computation an error is only uncomfortable while
it is being made. Thereafter it is neither
favored nor unfavored compared to a correct
digit. This is why errors are difficult to
remove in a simple Brownian copying process.
CC
C
CE
EC
E
EE
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CC
Proofreading scheme by contrast behaves
kinetically like an energetically nondegenerate
case of simple Brownian copying, in which errors
are permanently uncomfortable, being hard to make
but easy to unmake, as if they had an intrinsic
energy cost even after incorporation. (This
would be the case if the original and copy
strands stuck together after copying.)
Exonuclease
Polymerase
C
CE
EC
E
EE
But energetically there is a difference. An
energetically nondegenerate scheme could maintain
an error probability lt ½ without dissipation at
zero drift, but the proofreading scheme requires
continuing dissipation, because of the cycling
action of polymerase and exonuclease undoing each
others work.
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One stage proofreading scheme can reduce the
error to as low as s2, in the limit of high
dissipation. A 2-stage generalization might
intersperse a provisional state (yellow) between
each stage of copying. A multi-stage
generalization would intersperse a chain of
N-1 provisional states, like a fractional still
or isotope en- richment cascade. We are
studying the Speed/Error/Dissipation tradeoff
for Brownian copying by efficient schemes of
this sort.
CC
CC
C
CE
C
CE
EC
E
EC
E
EE
EE
Another open problem is how to generalize this
analysis to more general kinds of computation
than simple tape copying.
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