Biomedical Engineering Reference
In-Depth Information
Arrhenius divided molecules in a chemical system into two types: normal and
activated. Activated molecules have an energy equal to
E
, the activation energy,
and participate in chemical reactions, while normal molecules are dormant. He
then postulated that the normal and activated molecules are in thermodynamic
equilibrium (43,44). Although the essence of Arrhenius' argument is correct, it is
now well established that Eq. [6] is actually probabilistic, derivable via statisti-
cal mechanical arguments. Modern kinetic theory holds that reactions can take
place only if molecules collide with an energy greater than or equal to some
minimum,
E
. Treating molecules as billiard balls, the kinetic energy of a colli-
sion increases linearly with the masses of the molecules and as the square of the
velocity of the molecules relative to one another. Using Maxwell-Boltzmann
theory, a velocity distribution function for the molecules in a system can be de-
rived, and the probability of a collision occurring with an energy greater than or
equal to
E
can be determined (45-47). The leading-order solution for the prob-
ability is proportional to the Boltzmann factor,
e
-
E
/
kT
, which can be thought of as
proportional to the fraction of molecules with energy,
E
, at a given temperature,
T
. Hence, the increase in the average energy per collision, and therefore, the
increase in chemical reaction rates as a function of
T
, is proportional to the
Boltzmann factor, which is precisely the solution to the Arrhenius equation, Eq.
[6]:
R
e
-E/kT
. It is important to note that temperature also increases the fre-
quency of collisions. However, the effect of this on reaction rates is a pre-factor
that scales slowly as a power of
T
, and, therefore, is subdominant to the expo-
nential behavior of the Boltzmann factor.
Biochemical reactions necessary for metabolism within organisms are simi-
lar to reactions in a chemistry laboratory, except enzymes catalyze many meta-
bolic reactions and the medium for the reactions is the mitochondrial membrane
immersed in water. For every species of organism, there is a minimum tempera-
ture below which metabolic rate ceases, an optimal temperature at which meta-
bolic rate is maximum, and a very narrow temperature range above the optimum
where metabolic rate rapidly decreases. We are primarily interested in the "bio-
logically relevant" temperature range defined to be between the minimum and
the optimum temperatures, where effects such as the freezing of water at low
temperatures or the denaturing of proteins at high temperatures are negligible
(1,17,18). If temperature is in a "biologically relevant" regime and the effects of
temperature on enzyme functions are sub-exponential, the temperature depend-
ence of metabolic reaction rates is the same as that of non-biological reactions,
so that metabolic rate scales with a Boltzmann factor,
B
e
-E/kT
, where
E
now
represents an activation energy for metabolic reactions (1,46). When combined
with body size scaling, this implies that the characteristic power per gram, and
thus, typical biological rates (1,5) scale as
R
BIO
M
-1/4
E
-E/kT
,
[7]
