Screening of Electrostatic Interactions Screening of Electrostatic Interactions

Interactions between charges relevant in biology are almost always affected - and usually reduced in strength or screened - by the presence of many nearby water molecules, ions, and other molecules.


Unscreened (Vacuum) Coulomb Interaction

We all know that in vacuum, two charges q1 and q2 separated by a distance r have a Coulomb interaction energy of

U(r) = kq1 q2
r
where k = 1/(4pe0) = 8.99 ×109 N·m2/C2.

We call this a long-ranged interaction because of its slow power-law decay. In vacuum, the field lines starting out from an isolated charge flow off to infinity without terminating.

Essentially all interactions relevant to biology boil down to superposition of a lot of fundamental Coulomb interactions. But it is extremely impractical to think in terms of fundamental interactions when thinking about molecules in liquid water.


Dipoles in Polarizable Media Reduce The Strength of the Coulomb Interaction

We learned in electrostatics that charges embedded in a dielectric material interact by a Coulomb interaction which is reduced in strength by a dimensionless factor called the dielectric constant e:

U(r) = k q1 q2
er
This reduction in strength of the Coulomb interaction is due to the polarization of the particles of the dielectric medium - either induced or permanent dipoles around a free charge will be oriented so as to terminate some of the field lines coming from the free charge.

Note that the Coulomb interaction continues to have its long-ranged character, just with a reduced strength. This is sometimes referred to as dielectric screening of the charge.

For pure water at room pressure and 25 C, e = 78.5, i.e. Coulomb interactions are reduced in strength by a factor of about 80 relative to the vacuum. This is because of the permanent dipole moment carried by every water molecule. Note that the electrons tend to stay close to the oxygen, leaving two positively charged protons on one side of an H2O.

Sometimes people refer to this reduction of the strength of the Coulomb interaction as `screening' of the charge.


Example: Estimate the range at which two electron charges have an interaction energy kB T, in pure water at standard lab pressure and temperature.

kB T = k q1 q2 / (er) or r = k e2 / (ekB T)

Plug in k = 9×109 N·m2/C2, e = 1.6×10-19 C, e = 80, and kB T = 4×10-21 J, to find r = 7×10-10 m = 7 Å

This distance is often called the Bjerrum length

lB = k e2
ekB T
=     7    Å =     0.7     nm


Concentration and Molarity

We will soon be talking about concentrations a lot, and we will refer to concentrations in two different ways.

Number density - we'll use the symbol r to refer to number of molecules per volume, either in m-3 or in cm-3.

Molarity - to read the biochemical literature you need to understand molar concentrations. The molarity of X is often expressed as [X], and is simply the number of moles per litre.

A mole is Avogadro's number (NA = 6.02×1023) molecules.

You should remember that a liter is the volume of a cube which is 10 cm on a side, and therefore that 1 liter = 1000 cm3 = 10-3 m3.

You may also recall that a litre of water at room temperature/pressure has a mass of 1 kg.

Finally you should remember that the chemical weights given for elements in the periodic table, and on the sides of jars of chemicals, are the molar masses in g, i.e. the masses of NA = 6.02×1023 molecules. Thus the mass of a mole of hydrogen atoms is close to 1 g; the mass of a mole of O2 molecules is about 32 g (2 times 16 g).


Example: Find the molarity of water, for pure water.

We first note that a mole of water molecules has a mass of about 18 g.

So, [H2O] = number of moles in a litre of water = (1000 g) / (18 g) = 55.

We say that pure water is at 55 M concentration, in pure water.


Example: Find the number density, and estimate the mean distance between molecules, which are at 1 M concentration.

1 M means 6.02×1023 molecules/liter, or a number density of r = 6.02×1023 / (103     cm3) = 6 ×1020/cm3.

The average distance between molecules is just the 1/3 power of the volume per molecule, or
distance = [1/(6.02×1020     cm-3]1/3 = 1.2 ×10-7 cm
or about 1.2 nm or 12 Å.

Once you know this, you can quickly estimate the distance between molecules in solution at some other concentration (note the 1/3 power!)

at 10-3 M, the mean distance between solute molecules is 12 nm;

at 10-6 M (micromolar) the mean distance between solute molecules is 120 nm;

and at 10-9 M (nanomolar or nM) the mean distance between solute molecules is 1.2 microns.

A final note - concentrations in molecular biology are also often described using mass of solute per volume, usually in g/l (grams per litre) = mg/ml (milligrams per millilitre) = mg/cm3

= mg/ml (micrograms per microlitre).
Conversion from molarity to mg/ml obviously requires you to know the molar mass of the solute.


Example: How many g of NaCl should be added to 1 l of pure H2O to prepare a 100 mM concentration of Na+?

Easy - we need to put 0.1 moles of NaCl into a litre of water. Molar mass of NaCl = 23 + 35.5 g = 58.5, so we should add 5.85 g of NaCl to 1 l of water.

Sounds easy, but people in labs screw this kind of thing up all the time!


Free Ions Strongly Screen Coulomb Interactions

All aqueous solutions - even pure water - contain ions. Those ions can terminate electric field lines, and therefore can severely screen Coulomb interactions of charges. Important examples of ions relevant to biology:

Univalent ions - the cations Na+ and K+ are present at roughly 0.1 M concentrations, outside and inside cells, respectively. There are negative ions (`anions' or `counterions') at the same concentration to balance the charge; in the biochem lab this is often Cl-, and in the cell most of these `counterions' are glutamate ions.

Divalent ions - charge-2 cations like Mg2+ and Ca2+ are present at roughly mM concentrations in cells, and in many biochemistry expeiments.

Charged molecules - many proteins, nucleic acids, and other organic molecules in cells are charged, i.e. they give up ions to solution when they are put in water. A good example is DNA, which has one phosphate ion (PO4-) on each nucleotide. The counterion is usually Na+.

Water itself - pure H2O has pH 7.0, which means that the concentrations of hydronium (protons) and hydroxyl ions are [H+] = [OH-] = 10-7 M. So - even pure water is not the simple dielectric of elementary electrostatics (in general, nothing about water is simple).

Free ions do more than reduce the overall amplitude of Coulomb interactions - they change the shape of the potential energy, making it go to zero exponentially (rapidly) beyond a characteristic distance called the Debye screening length.

Roughly speaking, in solution with ions present, the 1/r Coulomb interaction is modified to have the screened Coulomb form:

U(r) =









k q1 q2
er
            short     distances

kq1 q2
er
e-r/lD     long     distances

Here, small and big r are roughly the cases where there are, respectively, no charges, and many charges between q1 and q2. When there are no ions between q1 and q2, the interaction is the usual 1/r potential. But when you separate the two charges to a sufficient distance that in a volume of diameter r you have many ions, those ions will organize so as to terminate the field lines of q1 and q2, thus eliminating their long-ranged interaction.

Unfortunately the theory behind this is a bit hard - in most cases of interest, the effective coupling k is exceedingly difficult to calculate. And the general form of the interaction in the middle-range where there are only a few ions in the volume between q1 and q2 is in general not too well understood, especially in cases where divalent, or worse, multivalent ions are present.

But - the main point is that as long as there are many ions between two charges, their interaction is screened strongly, simply because the ions can terminate electric field lines. A free ion attracts ions from solution of opposite sign, making a little `counterion cloud' which neutralizes its charge, and therefore by Gauss's law, basically eliminates the electric field.

The size of this `cloud' is roughly the screening length lD, the parameter that determines when the exponential `cuts off' the Coulomb interaction in U(r). A useful formula for lD is due to Debye, which comes from a certain relatively-easy-to-solve limiting case of interaction of charges with free ions present:

lD =



ekB T
4pk e2

i 
ri zi2




1/2


 
where the sum over i is over all the types (species) of ions, and where ri and zi are the number densities and valencies of the various types of ions. As you can see, as you add more and more ions, because the valences enter squared, the screening length goes down, down, down.

This formula is often called the Debye screening length, and provides a good first estimate of the distance beyond which Coulomb interactions can be essentially ignored, as well as the size of the region near a point charge where opposite-charge counterions can be found.

For aqueous (water) solution at room temperature, it is handy to rewrite the Debye screening length in terms of the Bjerrum length,

lD = 1

4plB

i 
zi2 ri
1/2
 
where you recall that lB = 0.7 nm.


Example: What is the Debye screening length in 1M NaCl aqueous solution?

Well, above we figured out that r = 6.02 ×1020     cm3 for 1 M concentration, so both Na+ and Cl- are present at this number density. Their valences are z = +1 and -1 respectively, giving us
lD = 1 / [4p×0.7×10-7     cm ×(6.02 ×1020 + 6.02 ×1020) cm-3]1/2

= 0.3 ×10-7 cm
or lD = 0.3 nm for a 1 M 1:1 electrolyte (in this case, Na+:Cl-).

The point is that for 1 M 1:1 ionic solution you have a screening length of less than 1 nm, meaning that at even a couple of nanometers separation, two charges no longer appreciably interact by the Coulomb interaction.


This example leads to three handy formulae for the Debye screening length for common ionic conditions at room temperature:

lD = [0.30     nm/[NaCl] ] for 1:1 electrolytes (e.g. Na+:Cl-)

lD = [0.18     nm/([MgCl2] )] for 2:1 electrolytes (e.g. Mg2+:2Cl-)

lD = [0.15     nm/([MgSO4] )] for 2:2 electrolytes (e.g. Mg2+:SO42-)

These formulae is stolen from Israelachvili's book (see references below) and are incredibly useful.


Inside cells where there is always > 150 mM of univalent salt around, this means that Coulomb interactions have a range of about 1 nm (=0.3/[0.15] nm)

Note that even pure deionized distilled water has a not-too-long screening length, since there is 107 M concentration of H+ and OH- ions excited thermally (we say that the pH of pure distilled deionized water is 7). The screening length in this case - the maximum possible in water - is 0.3/ [(10-7)] nm 1000 nm = 1 micron. So for separations beyond a few microns, even in absolutely pure water, two ions no longer `see' one another via the Coulomb interaction.


Charged surfaces trap counterions

The surfaces of large proteins, nucleic acids, cell membranes, and many other surfaces relevant to biology, are often charged. The charges are often important for solubulizing the proteins or membranes (as we have already mentioned for DNA). In any case, those charged surfaces, when immersed in solution where ions are present, will attract a thin `atmosphere' of opposite-charge counterions.

Of course, the thickness of this charge layer is about lD thick. The resulting sandwich of opposite-sign charges is often called an electric double layer.

As you might guess, an implication of this is that charged surfaces, and therefore biomolecules, only interact by Coulomb interactions when they are less than a few lD from one another. Under than a few nm from one another to `feel' one another, and to interact.

This leads to perhaps the only thing that is a simplification of molecular biology relative to chemical engineering - in general we can think of the Coulomb interaction as being a short-ranged interaction, or even a `contact' interaction.


Further reading

A superb introduction to interactions between molecules, and with some attention paid to colloids and biology is Intermolecular and Surface Forces with Applications to Colloidal and Biological Systems, J.N. Israelachvili, Academic Press 1985 (there is probably a newer edition, but the original version is one of my favorite books).


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On 1 Feb 2001, 13:10.