7.2. Osmotic pressure[[Student version, January 17, 2003]] 221
The osmotic pressure needed to keep our protein solution inside a cell, with pure water outside,
is thenkBTrc≈ 300 Pa.That’s certainly much smaller than atmospheric pressure (10^5 Pa), but is
it big for a cell?
Suppose the cell has radiusR=10μm.Ifit’s under 300Paof excess internal pressure, this
will stretch the membrane with a certain tension. Surface tension means that every part of the
membrane pulls on every other part. We describe it by imagining a line drawn on the surface;
the membrane to the left of the line pulls the membrane to the right with a certain force per unit
length, called thesurface tensionΣ. But force per length has the same units as energy per area,
and indeed to stretch a membrane to greater area, fromAtoA+dA,wemust do work. If we
draw two closely spaced, parallel lines of length ,the work to increase their separation fromxto
x+dxequals ( Σ)×dx. Equivalently, the work equals Σ×dA,where dA= dxis the change
inarea.Similarly, to stretch a spherical cell from radiusRtoR+dRwould increase its area by
dA=(dR)ddAR=8πRdRand cost energy equal to Σ×dA.
The cell will stretch until the energy cost of further stretching its membrane balances the free
energy reduction from letting the pressurized interior expand. The latter is justpdV=pddVRdR=
p 4 πR^2 dR.Balancing this gain against Σ× 8 πRdRshows that the equilibrium surface tension is:
Σ=Rp/ 2. Laplace’s formula (7.9)Substituting our estimate forpyields Σ = 10−^5 m· 300 Pa/2=1. 5 · 10 −^3 N/m.This tension turns
out to be roughly enough to rupture a eukaryotic cell membrane, destroying the cell. Osmotic
pressure is significant for cells.
The situation is even more serious with a small solute like salt. Bilayer membranes are only
slightly permeable to sodium and chloride ions. A 1Msalt solution contains about 10^27 ions perm^3 ,
ten thousand times more than in the protein example above! And indeed it’s well known that you
cannot dilute red blood cells with pure water; at low concentrations of exterior salt they burst, or
lyse.Clearly to escape lysis, living cells must precisely fine-tune their concentrations of dissolved
solutes, an observation to which we will return in Chapter 11.
7.2.2 Osmotic pressure creates a depletion force between large molecules
TakealookatFigure 7.2. One thing is clear from this picture: It’s crowded in cells. Not only
that, but there is ahierarchyof objects of all different sizes, from the enormous ribosomes on down
to sugars and tiny single ions (see Figure 2.4 on page 33). This hierarchy can lead to a surprising
entropic effect, called thedepletion interactionormolecular crowding.
Consider two large solid objects (“sheep”) in a bath containing a suspension of many smaller
objects (“sheepdogs”) with number densityc.(Admittedly it’s an unusual farm where the sheepdogs
outnumber the sheep.) We will see that the sheepdogs give rise to an effect tending to herd the
sheep together, a purely entropic force having nothing to do with any direct attraction between the
large objects.
The key observation, made by S. Asakura and F. Oosawa in 1954, is that each of the large
objects is surrounded by adepletion zoneof thickness equal to the radiusaof the small particles;
the centers of the small particles cannot enter this zone (Figure 7.3). The depletion zone reduces
the volume available to the small particles; conversely, eliminating it would increase their entropy
and hence lower their free energy.