Which factors determine the rate of movement of molecules across the membrane?

Since the net movement of diffusing molecules depends on the concentration gradient, the rate of diffusion is directly proportional to the concentration gradient (dC/dx) across the membrane. The concentration gradient, dC/dx, is the difference in molecule concentration inside and outside of the cell across a cell membrane of width dx. This is equivalent to (Cout - Cin)/Dx where Cout and Cin are the substrate concentrations inside and outside the cell, and Dx is the width of the cell membrane. When the concentration outside the cell (Cout) is larger than inside the cell (Cin), the concentration gradient (dC/dx) will be positive, and net movement will be into the cell (positive value of dn/dt).

We can describe the rate of diffusion as directly proportional to the concentration gradient by the following equation:

where A is the membrane area and P is the permeability constant. P is a constant relating the ease of entry of a molecule into the cell depending on the molecule's size and lipid solubility.

Notice that when A and P are constants, this equation simply describes a line where dn/dt is a function of dC/dx. If we graph the rate of diffusion as a function of the concentration gradient, we get a simple linear function.

Interpretation:

Notice the rate of diffusion increases as the concentration gradient increases. If the concentration of molecules outside the cell is very high relative to the internal cell concentration, the rate of diffusion will also be high. If the internal and external concentrations are similar (low concentration gradient) the rate of diffusion will be low.

Facilitated Diffusion

Variables:

n

number of molecules inside cell

t

time (seconds)

Vmax

saturation constant (mol/cm3/sec)

K

constant determining speed of saturation (mol/cm4)

C

concentration of diffusing molecule (mol/cm3)

x

width of cell membrane (cm)

Method:

Unlike simple diffusion, facilitated diffusion involves a limited number of carrier proteins. At low concentrations, molecules pass through the carrier proteins in a way similar to that of simple diffusion. At high solute concentrations, however, all the proteins are occupied with the diffusing molecules. Increasing the solute concentration further will not change the rate of diffusion. In other words, there is some maximum rate of diffusion (Vmax) when all the carrier pro teins are saturated. Therefore, we can not use a simple linear equation to describe the rate of diffusion. The rate of diffusion will increase with increasing solute concentration, but must asymptotically approach the saturation rate, Vmax. How quick ly the carrier proteins become saturated can be described by the variable K, the concentration gradient at which the rate of diffusion is 1/2 Vmax. K and Vmax depend on properties of the diffusing molecule, such as its permeability (P), as well as the surface area (A) of the cell, but for simplification we give the equation as:

We can graph this equation, dn/dt as a function of dC/dx, to see how the rate of diffusion changes with increasing solute concentration outside the cell.

Interpretation:

By graphing this equation, we see that at low concentrations of solute, the rate of diffusion into a cell occurs almost linearly, like simple diffusion. Notice that at low solute concentrations, the slope is much steeper than that of simple diffusion. Facilitated diffusion can increase the rate of diffusion of particular molecules at low concentrations. However, the rate of facilitated diffusion levels off with increasing solute concentration. Additional increases in external solute concentration cannot increase the rate of diffusion once carrier proteins are saturated.

Conclusion:

Passive diffusion of solute into a cell is linearly related to the concentration of solute outside the cell. Carrier proteins increase the rate of diffusion by allowing more solute to enter the cell. Facilitated diffusion, however, approaches a maximum rate as the carrier proteins become saturated with solute.

Additional Questions:

Simple Diffusion

1. We graphed dn/dt as a function of dC/dx. What is the slope of this line? What do increases or decreases in the slope mean biologically?

2. Now assume the concentration gradient is a constant. How does the rate of diffusion (dn/dt) change with the surface area (A) of the cell and the permeability (P) of the diffusing molecule? Graph dn/dt as a function of A or P and describe the function.

Facilitated diffusion

1. Look at the equation for facilitated diffusion and find the horizontal asymptote. What happens to dn/dt as dC/dx approaches infinity?

2. Try graphing this equation with different values for K. How does this change the concentration at which the carrier proteins are saturated?

3. Compare simple and facilitated diffusion of glucose into erythrocytes by graphing rate of diffusion (micromoles per hour) as a function of external glucose concentration (mmol/cm3). For facilitated diffusion, Vmax=500 micromoles per hour and K=1.5 mmol/cm3. For simple diffusion, A x P is 3 cm3/hour.

Sources:

Darnell, J., H. Lodish, and D. Baltimore. 1986. Molecular Cell Biology. Scientific American Books, Inc., New York

LeFevre, P. G. 1961. Sugar transport in the red blood cell: Structure-activity relationships in substrates and antagonists. Pharmacological Review 13:39-70.

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