Clas Blomberg - Physics of life-Elsevier Science (2007)

Chapter 30. Recognition and selection in biological synthesis 325
We can give some numbers to demonstrate the principles. In order not to work with too
large or too small numbers, let the Boltzmann factor that provides an equilibrium accuracy
be 100. This is smaller than the situation for DNA, but reasonable for the later examples.
A testing step, where a certain binding is relevant, may be close to equilibrium, which
would mean that 100 times more correct units are accepted than non-correct ones.
To improve this, there can be a proofreading step that may reject previously accepted bases.
This rejection step may again be 100 times faster for the non-correct bases. In the best possible
case, this might mean that the accuracy increases to a factor 100 100 10,000, which
means that there is one wrong unit accepted per 10,000 correct ones. The ratio of rejected
units depends on the ratio between the rejection rate and a rate to continue to the next state that
might mean final acceptance. Let us assume that a factor 1/(n 1) of correct bases are
rejected, n/(n 1) accepted. If the rejection rate is 100 times larger for the wrong units, the
corresponding factors become: a factor 100/(n 100) of incorrect units are rejected, a factor
n/(n 100) are accepted. Then, if n 1, one of every two correct units are rejected and
a factor 2/101 of non-correct ones are accepted. The total accuracy (correct units accepted/
non-correct units accepted) would be 10,000/2 5000, a factor 2 worse than the maximum
value. If n 10, most correct bases are accepted, only 1 of 11 are rejected, but a factor
10/110 of non-correct units are accepted. This would mean that the total accuracy would be
only 1000 instead of the best possible, 10,000. On the other hand, if n 0.1, then most
units are rejected, only 1 of 11 of correct ones is accepted, and only one of 1001 incorrect
units is accepted. The total accuracy 100 (1001/11) 9100, almost the maximum possible
value (10,000). There is a general conclusion here: to get a high accuracy, most correct
units have to be rejected in the proofreading step. It means a considerable free energy
cost as the rejected units are of a low-energy form and the original free energy of the insertion
reaction is lost as dissipation, not used for any work. (Well, it is used for improving the
accuracy.)
There is a thermodynamic cost of the high accuracy improvement by proofreading. (An
accuracy equal to what is provided by the Boltzmann factor can be obtained in equilibrium,
which needs no dissipation.)
There is another cost: the testing takes time, and this is so also for the simple selection
process without proofreading. Of course, the proofreading step implies a delay: a number
of correctly accepted units are rejected and must re-enter the selection processes. In a first
selection step a unit is bound by an association rate and may then dissociate with a rate that
is faster for an incorrect unit. This yields a main first testing, and for this, the primary
bound complex should be close to equilibrium, and the dissociation rate should be relatively
rapid. Thus, a good testing means a long testing time.
A relevant factor for the proofreading is the energy level of the rejected unit. The possibility
for the low-energy units to go back and enter the selection process through that step
should be improbable. But how improbable? Well, the aim of the proofreading is to enhance
an initial accuracy, and this requires of course that there is an outgoing, rejecting flow
through the proofreading step. It is necessary that there is a free energy decrease through
that step. This requires that the free energy (chemical potential) of the rejected states must
be lower than the free energy (chemical potential) of the units that pass the first initial step.
For DNA, a very high accuracy is needed, and the initial step can accomplish a fairly high primary
accuracy, say about 1 incorrect per 10,000–100,000 correct units, the final accuracy