Biomedical Engineering Reference
In-Depth Information
whereas typical times (5) scale as
t
BIO
M
1/4
e
E/kT
.
[8]
This exponential dependence on temperature is closely tied to
Q
10
factors,
which are frequently used in biology. A
Q
10
factor is defined as the percentage
change in a biological rate or time when there is a 10(C change in temperature.
Since the temperature in degrees Celsius is given by
T
c
=
T
- 273, it follows that
to leading order,
e
-
E
/
kT
e
B
T
c
, where B =
E
/
k
(273)
2
. A value of
E
0.6 eV is typi-
cal for metabolic reactions, so B 0.1(C
-1
. Thus, a typical value for
Q
10
, which
is the ratio of rates, is
Q
10
e
2.72. Since
Q
10
factors are only approximations
to the exact formula, they are probably best used as a guideline, and Boltzmann
factors should be used for precise tests or calculations.
The activation energy,
E
, can be interpreted in a couple of ways, depending
on whether the core reactions for metabolism occur in series or in parallel. If the
reactions occur in series, the rate of supply of reactants to a reaction is set by the
preceding reaction in the chain, and thus, the overall rate of the chain of reac-
tions is set by its slowest, rate-limiting step. In this case,
E
is simply the activa-
tion energy of this rate-limiting reaction. However, if there are
N
core reactions
that occur in parallel and are supplied with molecules from different pools, the
overall rate of metabolism is given by the average rate, which to leading order is
proportional to
1
N
.
e
E
/
kT
[9]
i
N
i
=
0
In the special case where t
he
activat
io
n energie
s
for all of the core reactions are
near some average value,
E
(
E
i
=
E
or (
E
i
-
E
)/
k
T
<< 1 for all
i
), the overall
reaction rate for metabolism is approximately
e
. Similarly, if there i
s
a
subset of reactions that have activation energies near some average value,
E
,
and this average is significantly sma
l
ler than activation energies for all other
reactions, the rate will be given by
E
/
kT
e
. Real biochemical networks consist of
a combination of reactions in series and reactions in parallel. Since many em-
pirical results agree with the temperature dependence in Eqs. [7] and [8] (see the
ยง1), it is likely that one of the scenarios outlined above, or some combination
thereof, is correct for biological systems.
For a wide range of rates and times across a broad assortment of taxa,
measured values of
E
are typically in the range of 0.6-0.7 eV, reflecting some
shared biochemistry (1,5,17,18). Indeed, these values for
E
bracket the average
value for biochemical reactions (48,49) and are very close to the activation en-
ergy for the oxidation of NaDH, which is common to most of life and may be a
rate-limiting step!
E
/
kT
