Biological Physics: Energy, Information, Life

7.4. A repulsive interlude[[Student version, January 17, 2003]] 229

have pure water on both sides of the membrane, there will be flow if wepushon one piston. Since
the pores are generally small and the flow slow, we expect a Darcy-type law for this “hydraulic
permeation” (see Section 5.3.4 on page 159). If there is a fixed density of pores per unit area,
weexpect a volume flow (volume per time) proportional to the applied pressure and to the area.
The corresponding volume flux is thenjv=−Lp∆p,whereLpis a constant called thefiltration
coefficientof the membrane (see Problem 4.10 and Section 5.3.4 on page 159). The discussion
above suggests that there is a generalization of the hydraulic permeation relation to embraceboth
driven and osmotic flow:^4

jv=−Lp

(

∆p−(∆c)kBT

)

. volume flux through a semipermeable membrane (7.15)

Equation 7.15 establishes a quantitative link between driven permeation and osmotic flow, two
seemingly different phenomena. If we apply zero external force, then osmotic flow proceeds at a
ratejv=LpkBT∆c.This is the rate at which the entropic force per area, (∆c)kBT,just balances
the frictional drag per area,jv/Lp.Asweincrease the opposing applied pressure the volume flux
slows, drops to zero when ∆p=(∆c)kBT,thenreversesat still greater ∆p,giving reverse osmosis.
Equation 7.15 actually transcends the rather literal model of a membrane as a hard wall pierced
with cylindrical channels, introduced above for concreteness. It is similar in spirit to the Einstein
relation (Equation 4.15 on page 108), as we see from the telltale presence ofkBTlinking mechanically
driven to entropically driven transport processes.
T 2 Section 7.3.1′ on page 249 mentions the more general situation of a membrane with some
permeability tobothwater and dissolved solute.

7.4 A repulsive interlude

Until now we have studied osmotic forces under the assumption that interactions between solute
particles can be neglected. That may be reasonable for sugar, whose molecules are uncharged, but
as we’ll see in a moment, electrostatic interactions between the objects contained in a cell can be
immense. Accordingly, this section will introducemixedforces, those that are partly entropic and
partly energetic.

7.4.1 Electrostatic interactions are crucial for proper cell functioning

Biomembranes and other big objects (such as DNA) are often said to be “electrically charged.”
The phrase can cause confusion. Doesn’t matter have to be neutral? Let’s recall why people said
that in first-year physics.

Example

Consider a raindrop of radiusr=1mmsuspended in air. How much work would

beneeded to remove just one electron from just 1% of the water molecules in the

drop?

Solution: Removing an electron leaves some water molecules electrically charged.

These charged water molecules crowd to the surface of the drop to get away from

each other, forming a shell of charge of radiusr.Recall from first-year physics that

(^4) Some authors introduce the abbreviation Π =ckBTwhen writing this formula, and call Π the “osmotic pressure.”
Some even regard osmotic pressure as a fictitious agent exerting an extra mechanical force. We will avoid these
confusing locutions and simply call this quantityckBT.