constants of the reactions. Assuming that k 3 is much smaller than k 1 , the transport velocity (v), or influx,
is given by

v
k 3 [ES] (13)

Maximal transport velocity Vmaxshould be attained at saturating solute concentration, when all carrier
molecules are occupied by the solute:

Vmax
k 3 [ET] (14)

where [ET] is the sum of occupied and unoccupied carriers or the total carrier concentration ([ ET]
[E]
[ES]). The affinity of the carrier for the solute is the reciprocal of the dissociation constant of the car-
rier-solute complex or the reciprocal of the Michaelis-Menten constant (Km):

Km


k

k

2

1




[E

[E

][

S

S

]

o]

 (15)

KMandVmaxcustomarily define transport kinetics. To calculate the transport velocity, [ E] in Eq. (15) is
replaced by ([ ET] [ ES]), and the equation is rearranged:

Km


([ ET]

[E

[

S

E

]

S])[So]




[E

[

T

E

]

S

[S

]

o]

[So] and [ ES]
K

[

m

ET



][S

[S

o]

o]

The latter equation can be substituted for [ ES] in Eq. (13), that is, v
k 3 [ET][S]/(KM[S]), and by re-
placing (k 3 [ET]) with Vmax[ Eq. (14)] the Michaelis-Menten equation is obtained:

v


K

V

m

m



ax[S

[S

]

]

 (16)

This equation can be transformed into a linear function of the reciprocals of v(same as Jo→i) and S(same
asajo):



1

v
Vm

K

ax

M

[S]



Vm

1

ax

 (17)

whereKM/Vmaxis the slope of the line and 1/Vmaxis the Y-axis intercept. Substitution of Vmax/ 2 for v
shows that KMis the solute activity when the velocity is half-maximal. Various graphic analyses [84]
show that competitive inhibition (inhibition by competition for carrier sites) of transport increases the ap-
parentKM(decreases apparent affinity) but does not affect Vmax.
Strict compliance with Michaelis-Menten kinetics is usually obtained only in a narrow solute con-
centration range and, in particular, in the low concentration ranges up to about 0.7 mM. In broader con-
centration ranges, multiphasic uptake kinetics [85] are ordinarily encountered. The reason for multipha-
sic kinetics has not been resolved; various possible explanations have been discussed [85–90].

2. Primary Active Transport

The term primary active transport [7] is reserved for active transport that is directly driven by energy-rich
metabolites, such as ATP, pyrophosphate, or electron donors. Cotransport is classified as secondary ac-
tive because it derives its energy from the electrochemical potential difference that is produced by pri-
mary active transport.
Two types of adenosinetriphosphatase (ATPase), at the plasma membrane and the tonoplast, respec-
tively, and an inorganic pyrophosphatase (PPiase) at the tonoplast are known to generate an electro-
chemical proton potential difference, namely a proton motive force, at these membranes [32]. In addition,
a Ca^2 -ATPase performs primary active Ca^2 transport. Proton motive force is also generated by vecto-
rial electron transport across the inner membranes of mitochondria and chloroplasts. This proton motive
force is primarily used for ATP synthesis, catalyzed by a third type of ATPase (an ATP synthase) [91].

PROTON MOTIVE FORCE GENERATING ATPases The proton motive force-generating H-AT-
Pase at the plasma membrane transports protons actively from the cytoplasm to the free space. At the
tonoplast, the V-ATPase and the PPiase pump protons actively from the cytosol into the vacuole. Both
ATPases need Mg^2 to function; their substrate, indeed, is Mg-ATP. These ATPases differ in their evo-
lution, in their homology, and in some of their characteristics from bacterial and animal ATPases [92].

MINERAL NUTRIENT TRANSPORT IN PLANTS 345