In considering the transport of a particular species of molecule or ion, it is useful to start by constructing a

The relationship between the chemical potential of molecular species i and the variables that are important in considering transport processes is given by the following expression:

µ

where:

C is, in most cases of biological interest, the concentration in moles/liter. Actually it is the activity which should appear here, and sometimes this distinction is significant.

V is the electrical potential. It seldom appears except as a gradient or a difference in voltage between two regions.

P is the pressure.

µ

zF is the electrical charge per mole for a charged ion. z will often be a positive or negative integer, but in some situations it may be useful to use a fractional value representing a mixture of ionic species, rather than dealing with each species individually. Faraday's constant, F, is 96,500 Joules/volt, or 23.06 kcal/volt.

v is the partial molar volume.

That equation (1) does in fact constitute a potential function can be seen clearly from the fact that the last two terms in equation (1) give the work done per mole when molecules are moved between regions of different voltage or pressure. This may not be so obvious for the term involving concentrations, but this term is familiar from the usual relations between free energy changes and concentrations in chemical reactions.

For comparison with the free energy change values for chemical reactions it is useful to obtain numerical values for µ in units of kcal/mole, in which case equation (1) can be used in the following form (for a temperature of 25 C):

µ

where V is in volts, v

Since µ is a potential function, its first derivatives with respect to spatial coordinates represent forces tending to move the molecules to regions of lower potential, thus minimizing the total free energy of the system. This differentiation requires, of course, that the potential be a continuous function of position, which will normally be true in a homogeneous phase. At a phase boundary, or in other heterogeneous systems, the situation will be qualitatively similar, but other theoretical treatments may be necessary to obtain kinetic parameters. In a homogeneous phase,

Force/mole of the i

If the molecules on which this force acts are free to move, they will reach a steady

Drift velocity = -u

where the constant, u

Flux =

Since fluxes are normally expressed as moles/sec/cm

This gives us the standard form of

If voltage differences are the only source of forces acting on the molecules,

which is also familiar, although it is common to use a mobility constant which has units of cm/sec//volt/cm, in which case the factor z

The purpose of these arguments is to show that the basic flux equation, equation (5), is a reasonable generalization which includes the most familiar transport phenomena, so that it can be used as a definition of passive transport. Passive transport is transport which is driven by the gradient of chemical potential of the molecular species which is being transported. It is always in the direction of chemical equilibrium for this species; that is, it tends to level out any differences in chemical potential. It is always associated with a decrease in the total free energy of that species in the system under consideration.

Obviously, any transport that does not use only this source of energy to overcome the frictional resistances to movement must utilize some other source of energy, and if the transport occurs in opposition to a gradient of chemical potential, additional energy is required to increase the potential of the transported molecules. Such transport, brought about by energy derived from the metabolism of an organism, is referred to as

This distinction between active and passive transport is not completely unambiguous, because it has neglected the possibility of interactions between a flux of one kind of molecule and the other molecules in the solution, which may induce fluxes of these molecules which would not be expected on the basis of their electrochemical potential distributions. The appearance of such fluxes can be described by the term

At equilibrium there must be no net transport of components between the two compartments, and the electrochemical potential of any component that is free to move between the two compartments must be the same in both compartments.

µ

C

The ratio C

V

Or, from equation (2), the following form can be obtained which gives values in volts, for a temperature of 25 C.

V

The activity coefficient of the solute ion is assumed to be the same in both phases.

A concentration difference between the two compartments may be maintained at equilibrium if there is a pressure difference:

P

In most cases of biological interest, movement of solute molecules between the two compartments is restricted, but the solvent, water, can diffuse freely across the interface between the two compartments and must therefore satisfy the condition for equilibrium. Under these conditions the activity of water cannot be accurately expressed by its concentration in moles/liter, but must instead be expressed in terms of its mole fraction in the solution. Usually it is more convenient to have an expression for the pressure difference in terms of the solute concentrations, rather than the mole fraction of water. If the solute concentration is represented by c, the number of moles of water in one liter of solution will be given by

(1/v ) - kc

where kc represents the number of moles of water which must be missing from one liter of solution in order to accommodate c moles of solute. The mole fraction of water in the solution is

((1/v)-kc)/ ((1/v)-kc+c) = 1/( 1 + cv/(1-kcv))

In most situations of biological interest, where the solutions are relatively dilute, cv and kcv will both be small compared to one, since v for water is approximately 0.018 liters/mole. Thus approximately, the mole fraction is given by 1/(1+cv ) and its logarithm is approximately equal to -cv . Substituting this into equation (12) leads to the familiar expression for the osmotic pressure:

P

The symbol P is often used for osmotic pressure, to distinguish it from hydrostatic pressure.

The pressure must be greater on the side with the greater solute concentration.

The total concentration of osmotically active solute in the cell is then 5 + 25 mM = 30 mM. (This is sometimes referred to as 30 milliosmoles.) If the cell is surrounded by distilled water, there will be a strong driving force for water to enter the cell to dilute the solutes inside the cell. If there is a strong cell wall around the cell that can withstand this osmotic pressure, the cell will come to equilibrium with the pressure given by equation (13).

For a flexible animal cell, we will try to solve the problem by putting salt in the environment around the cell. Suppose the solution on the outside of the cell is a 15 mM solution of NaCl. The concentration of osmotically active solute will be 15 mM Na

If the cell membrane is permeable to Na

We now have a situation with a voltage across the membrane, with the outside + relative to the inside. This voltage difference will slow down the influx of Cl

[Na

We also have the equation for charge balance inside the cell:

[Na

From these equations, the concentrations of [Na

A more typical biological situation would be one where the external salt concentration is higher, say 0.15 M, as in typical vertebrate body fluids. In this case, the same procedure leads to the result that [Na

However, we can now see that this cell can achieve osmotic equilibrium if it can actively pump Na

The rate at which the active transport mechanism is required to pump Na