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

Chapter 30. Recognition and selection in biological synthesis 329
of amino acids are relevant in testing and also for time: Some amino acids occur much less
frequently than other and, as we have said, some are more difficult to distinguish. The
ratios of various amino acids are then quite relevant, and one can investigate what would be
most efficient for accuracy and process times. Evidently this is a problem that depends on
a number of factors and processes that influence each other. Some results were investigated
in von Heijne and Blomberg (1979) and Blomberg (1987).
30D
Formalism in non-branched processes without proofreading
We here show the formalism of the selection processes and relevant results. First consider
an unbranched process with a general scheme:
k1
S E ⇐⇒ ( ES) ⇐⇒( ES)
⇐⇒ P E
⇒
k1
k12
1 2
k21
k2
k2
k
(30.1)
As in other schemes the arrows with two ends mean that the reaction can go in both directions.
The upper symbol marks the rate towards the right, the lower the rate towards the left.
S marks substrates to be selected; E the selecting enzyme, (ES) i enzyme states, and P the
product. The last arrow means that the product is taken up in an irreversible way by further
processing. That may mean that it is strongly bound in some complex.
To treat this, it is further assumed that the substrate S has a constant concentration, which
can mean either that its concentration is large so that this reaction does not change its concentration
in a significant way, or that it is steadily produced in a controlled way, keeping
its concentration constant. What concerns amino acids, one can say that both arguments are
true, they are at high concentrations, but they are also all the time produced and kept at constant
concentrations. Amino acids are also liberated as proteins are broken down. We now
consider the scheme by formulas, and use the same notations for concentrations as for the
components; thus S stands for the substrate and for the concentration of the substrate (ES) 1
for the first enzyme-bound state and for its concentration.
The scheme can be formalised by considering flows through the states; the flow from the
first state to the second is J 01 k 1 E S k 1 (ES) 1 . We now assume that there is a constant
flow through all states. Apart from a possible initial stage, this is a reasonable assumption
for processes that go on all the time. This means:
J k 1 E S k 1 (ES) 1 k 12 (ES) 1 k 21 (ES) 2 k 2 (ES) 2 k 2 E P kP (30.2)
From these relations, one gets expressions:
( ES)
2
⎡ k Ek
⎢
⎣ k2
2
⎤
⎥
P
⎦