7.4. A repulsive interlude[[Student version, January 17, 2003]] 231
off with distance at all! In short,
Even though the electrostatic interaction is of long range in vacuum, in solution
the screening effect of counterions reduces itseffectiverange, typically to a
nanometer or so.(7.16)
We’d like to understand the formation of the counterion cloud, which is often called thediffuse
charge layer.Together with the charges left behind in the surface, it forms anelectrical double layer
surrounding a charged macroion. The previous paragraph makes it clear that the forces on charged
macroions have a mixed character: They are partly electrostatic, and partly entropic. Certainly if
wecould turn off thermal motion the diffuse layer would collapse back onto the macroion, leaving
it neutral, and there’d be no force at all; we’ll see this in the formulas we obtain for the forces.
Before we proceed to calculate properties of the diffuse charge layer, two remarks may help set
the biological context.
First, your cells are filled with macromolecules, but three quarters of your mass is water. A
number of attractive forces are constantly trying to stick the macromolecules together, for example
the depletion force, or the more complicated van der Waals force. It wouldn’t be nice if they just
acquiesced, clumping into a ball of sludge at the bottom of the cell with the water on top. The same
problem bedevils many industrial colloidal suspensions, for example paint. One way Nature, and
weits imitators, avoid this “clumping catastrophe” is to arrange for the colloidal particles to have
the same sign of net charge. Indeed most of the macromolecules in a cell are negatively charged
and hence repel each other.
Second, the fact that that electrostatic forces are effectively of short range in solution (summa-
rized in Idea 7.16 above) matters crucially for cells, because it means that:
- Macroions will not feel each other until they’re nearby, but
- Once theyarenearby, thedetailed surface patternof positive and negative residues on
aprotein can be felt by its neighbor, not just the overall charge.
As mentioned in Chapter 2, this observation goes to the heart of how cells organize their myriad
internal biochemical reactions. Though thousands of macromolecules may be wandering around
any particular location in the cell, typically only those with precisely matching shapes and charge
distributions will bind and interact. We can now see that the root of this amazing specificity is
that:
While each individual electrostatic interaction between matching charges is
rather weak (compared tokBTr), still the combined effect of many such in-
teractions can lead to strong binding of two molecules—iftheir shapes and
orientations match precisely.(7.17)
Notice that it’s not enough for two matching surfaces to come together; they must also be properly
oriented before they can bind. We say that macromolecular binding isstereospecific.
Thus understanding the very fact of molecular recognition, crucial for the operation of every
cell process, requires that we first work out the counterion cloud around a charged surface.
7.4.2 The Gauss Law
Before tackling statistical systems with mobile charged ions, let’s pause to review some ideas about
systems offixedcharges. We need to recall how a charge distribution gives rise to an electric field
E,inthe planar geometry shown in Figure 7.7. The figure represents a thin, negatively charged