
Chapter 7
Writing Rate Laws for Translocation Processes
Translocation is the third broad class of
biological processes. Since it includes all process that move
chemical species from one physical place to another, we need to
develop rate laws for processes as diverse as diffusion,
endocytosis, blood flow, and transport of mRNA from the nucleus
to the cytoplasm through the nuclear pore complex.
Diffusion is arguably the simplest of these
special cases. It is a process that occurs in inanimate systems
just as readily as it occurs in living systems. No specialized
proteins are required, unless the diffusing molecule must cross a
lipid membrane, and even then a nonpolar, lipid soluble molecule
can cross without proteinacious assistance.
Imagine a molecule of glucose in the
cytoplasm of a mammalian cell. In order for this molecule to
begin its trek through glycolysis, it must first be
phosphorylated. In light of what you learned in the previous
chapter, this phosphate transfer can only take place after the
glucose, the phosphoenolpyruvate, and the appropriate enzyme have
come together to form the enzymesubstrate complex. This coming
together is dependent on diffusion.
Diffusion is a special case of passive
movement in a chemical field. The chemical field is the force per
mole tending to drive the passive transport of a molecule from
one place (phase) to another. This field is conservative in the
sense that the free energy associated with a given position is
conserved: the work done in transporting a quantity of molecules
from one place to another is independent of the path taken. In
fact, any arrangement of stationary phases gives rise to a
chemical field that is conservative. The mathematical statement
of this physical fact is that the chemical field, F_{c},
can be written as the gradient (derivative with respect to space)
of the chemical potential.
Equation 1
If you are familiar with electromagnetic
fields, you will see that the chemical field is related to the
gradient of chemical potential just as an electric field is
related to the gradient of the electrical potential. An electric
field has units of N/coul (newtons per coulomb); a chemical field
has units of N/mole. For an uncharged molecule, s, the chemical
potential is given by
Equation 2
where m _{0} is a constant at a given temperature,
T, and pressure, P, R is the Gas Constant, C_{s} is the
concentration of s (molar), V_{p} is the partial molar
volume of s, and P is the pressure. Differentiating gives:
Equation 3
In respiratory and cardiovascular
physiology and in any experimental system using pumps, you will
encounter significant pressure gradients, but animal cell
membranes cannot support such gradients. Consequently, for
cellular level work, we will set dP/dx = 0 and the chemical field
will reduce to the concentration gradient term.
This force, F_{c}, tending to move
mass from an area of high chemical potential to an area of low
chemical potential is opposed by a friction force, F_{f},
proportional to the velocity of the transported mass:
Equation 4
where f is the coefficient of friction. You
may have encountered, in Physics, friction forces proportional to
v^{2}; these occur at velocities greater than those
encountered in biological systems. Particles of the substance, s,
will be accelerated by the chemical field until the field is
balanced by this friction force, that is until F_{c} = F_{f}.
Now we're getting close to the result we want, but to make it
useful the velocity, v, needs to be related to quantities we
actually measure. This is accomplished by recognizing that the flux,
J_{s}, of substance, s, (in moles sec^{1} cm^{2})
is equal to the product of concentration and velocity. Thus, the
friction force can be written as f J_{s}/Cs. Equating
this to our expression for the chemical field gives
Equation 5
With one last piece of information,
provided by Albert Einstein in 1905, this derivation is complete.
Einstein demonstrated that RT/f is equal to the classical
diffusion coefficient, D, used by Adolf Fick to characterize
diffusion 50 years earlier. This substitution yields Fick's Law
of Diffusion. Fick, however, did not derive his law from first
principles as we have done here. He did not even obtain the law
as an empirical fit to experimental data. Instead, he argued by
analogy to the movement of heat down temperature gradients, which
had been analyzed by Fourier in 1822. Thus, by calling equation 6
Equation 6
"Fick's Law of Diffusion", we are
honoring the intuition of a great scientist.
Pay attention to the units of flux. In
previous chapters we concluded that all of the differential
equations that we formulate for biological systems should be
written in terms of conservation of mass. You might expect,
therefore, that each term on the right hand side of your
differential equation would have units of mass/time. Fick's first
law, however, produces a flux whose units are, for example, nmol
sec^{1} cm^{2}. Indeed many scientists insist
that the word flux should only be used when the movement of
molecules is expressed in these terms. Rather than enter the
fray, I suggest that you listen carefully when "fluxes"
are discussed and be certain that you know which units are
intended. In many situations you will need to include a diffusive
flux term in a model you are constructing, but you will have no
estimate of the crosssectional area through which the flux is
travelling. The rate law can then be written to acknowledge that
the flux is proportional to the concentration difference,
Equation 7
where D' is related to the classical
diffusion coefficient but incorporates both the diffusional cross
sectional area and the thickness of the barrier separating the
compartments, 1 and 2. In this case, what are the units of D'? It
depends, of course, on what units you choose for [S]_{1}
and [S]_{2}; D' will have whatever units are required to
convert these concentrations to, say, nmol/sec. Notice that J_{diff}
changes sign if the concentration difference changes sign. This
reflects the passive nature of the process; at one moment the net
flux may be moving from 2 to 1 and the next moment from 1 to 2.
Why do we use the word "net" to
describe the diffusive flux? The answer is because there are
always molecules moving from compartment 2 to compartment 1 and
from compartment 1 to compartment 2. In fact, you could write the
rate law as which shows the individual forward and reverse
contributions to the net flux. You can see here that diffusion is
a bidirectional application of the principle of mass action. The
individual flux contributions are both proportional to the
concentration of S in the originating compartment, the
compartment from which the flux flows.
A final point, perhaps obvious by now, is
that the concentrations of S are absolutely required on
the right hand side of the equation. The mass of S cannot be
substituted for the concentration because you could then cause
transport up a concentration gradient just by giving the low
concentration compartment a much larger volume than the high
concentration compartment.
DIFFUSION COEFFICIENTS ARE ALMOST ALWAYS IN
THE RANGE FROM 106 TO 105 cm2/sec. In order to develop some
intuition about diffusion, you should learn the absolute values
of some typical diffusion coefficients. These are reported values
for diffusion in water at 20 oC. Notice that the bigger the
molecule, the smaller the diffusion coefficient.
PERMEABILITY CHARACTERIZES DIFFUSION
THROUGH MEMBRANES. To apply Fick's law to diffusion through a
cell membrane of finite thickness one must satisfy two
assumptions: 1) that the membrane thickness is small and 2) that
the inside and outside compartments are well mixed. When these
assumptions are made, the concentration gradient, dC/dx, can be
replaced with (CoCi)/b, where the subscripts o and i refer to
outside and inside and b is the thickness of the membrane. Then b
can be lumped into the diffusion coefficient and a new constant,
P = D/b, is formed. P is referred to as a permeability.
DIFFUSION COEFFICIENTS INCREASE WITH
INCREASING TEMPERATURE. This empirical fact was given a
theoretical basis in the Einstein relation (D = RT/f), mentioned
earlier. The temperature appears linearly in this equation, but
it is important to remember that this is absolute temperature.
Consequently, a change from 20 C to 30 C does not result in a 50%
increase in D. Remember that absolute zero is 273 C. Perhaps you
anticipated the increase in diffusion coefficient with
temperature since you know that increasing temperature causes
increased molecular velocity.
A quantity frequently used to
characterize the temperature sensitivity of a physical or
chemical process is the Q10. This is defined as the rate of the
process at T+10 C divided by the rate at temperature T. Certain
values of Q10 have sometimes been quoted as "proof"
that a process is energy requiring ("active") rather
than passive. Such arguments are not supported by the facts.
Activation energies for active and passive processes overlap
extensively. Another group of membrane proteins is comprised of
those that function as channels, carriers or ports. As indicated
by their names, these proteins mediate the transmembrane movement
of molecules that would otherwise not be able to pass through the
phospholipid bilayer. The three names suggest different
mechanisms of action, but, in fact, all of them include a
specific binding step (or steps) followed by translocation and
unbinding. In some cases, there are controls on the translocation
step such as gates in ion channels.
Another characteristic of these proteins, that you should always keep track of, is whether
or not they carry molecules up or down the chemical potential
gradient. If uphill transport is involved, there must be a source
of free energy (often an ion gradient produced by a pump) that is
being tapped to drive transport. The term pump is generally
restricted to those membrane proteins that carry out active
transport (that is, transport against a concentration gradient)
and which do so at the direct expense of ATP or some other high
energy phosphate. The increase in free energy that accompanies
transport up the concentration gradient is paid for by coupling
the process to the free energy supplied by ATP hydrolysis. The
overall free energy change is still negative, as it must be for
the pump to work.
All rate laws can be written at any of several
levels of detail. The simplest form of a rate law for any process
is the linear mass action rate law. This approach is frequently
taken for carriers, channels and pumps when electrical forces are
irrelevant or can be treated as constant for the duration of the
experiment. Ion channels rarely operate in a range in which they
saturate, but carriers and pumps saturate almost as regularly as
enzymes. It's not known why this is so, but it may depend on the
molecular mass of the ligands. If saturation is to be included,
formulate your rate law using the principles developed in the
chapter on enzyme catalyzed processes. The only difference is
that for a transport process, the substrate and product are
replaced by the transported species in two different physical
places. For example, if glucose (G) is the transported species
and it is transported across the cell membrane from extracellular
space to cytoplasm, then GECS goes into the diagram in place of S
(substrate) and Gcyto replaces P (product). Using the rapid
equilibrium approximation, the kinetic constants that appear in
the rate law will refer to equilibrium binding of G to the
extracellular face of the transporter and equilibrium binding of
G to the cytosolic face of the transporter. If the reverse flux
is neglected, then only the extracellular binding constant will
appear in the final rate law. This equivalence between transport
and enzymes is so widely recognized, that biochemists and
enzymologists often refer to transporters as
"permeases"  enzymes that catalyze permeation.
RATE LAWS FOR ION CHANNELS Ion channels may be treated as carriers if
the membrane potential can be treated as constant. Under these
conditions, the resulting rate law is simply linear mass action
because, as mentioned above, the saturation of ion channels is
only observed at very elevated, unphysiological, concentrations
of the permeating ion. Two other situations are commonly
encountered when modeling ion channels. First, the transmembrane
potential often changes. This may be physiologically induced or
be part of the protocol in voltageclamp experiments. In these
circumstances, you need a rate law that accounts for the
electrical contribution to the chemical potential because the ion
flux or current can be dramatically changed by changes in the
electrical driving force.
PASSIVE MOVEMENT OF IONS DEPENDS ON
BOTH CONCENTRATION AND ELECTRICAL POTENTIAL GRADIENTS. To obtain
an expression for the passive flux of any ion, we proceed as we
did for diffusion above. The net passive flux of the kth ionic
species can be written as where vk is the velocity and Ck is the
concentration of the species. In turn, the velocity may be
written as where uk is the mobility of the ion, and mk is its
chemical potential. This equation gives the velocity as the
product of velocityperunitforce (mobility) and force (gradient
of chemical potential).
In the last lecture we wrote the chemical
potential expression for an uncharged molecule; now we need the
corresponding equation for an ion. It is where z is the ionic
charge, F is the Faraday, V is the electrical potential and the
other terms are as before. Differentiating and substituting in
the equation for passive flux gives (pressure gradients are again
set to zero) or This is the NernstPlanck flux equation for the
passive movement of ions down an electrochemical gradient. Be
sure to distinguish the mobility from the chemical potential; uk
and mk are similar symbols, but they represent very different
quantities. The NernstPlanck equation is the rate law we
require. It expresses the flux of ions as a function of both
chemical and electrical driving forces.
One level more in sophistication of ion channel rate laws is often required if the
channel is gated by voltage or ligands. This topic is beyond the
scope of this small book and the interested reader is referred to
excellent books on the topic: Ionic Channels of Excitable
Membranes by Bertil Hille and references therein.
Chapter 8
 