1. Discussion topic for week 3 : Entropic forces
In the previous lectures we have stressed that random motion is the dominant feature of cellular dynamics.
Why don't cells exploit the long-range Coulomb interactions to accelerate enzyme reactions, binding of ligands to proteins, etc?

2. Thermodynamics in cells (Nelson, chap. 6)
See the Statistical Physics notes for detailed descriptions of Entropy,
Temperature and Free Energy.
Entropy is a measure of disorder in a closed system. It is defined as
where Ω is the number of available states in the phase space
Statistical postulate: An isolated system evolves to thermal equilibrium.
2nd law: DS  0, equilibrium is attained when the entropy is at maximum.
Entropy of an ideal gas:

3. Example:
Temperature is defined from entropy as
For the ideal gas:

4. Dx
To create order, we have to do
work on a system
volume change
work done
To push the piston f > FP, then DS > 0. At the equilibrium DS = 0

5. a
Open systems:
Consider the same system but now
immersed in a heat bath which
keeps the temp. constant at T.
As the spring expands
Consider the total entropy of the system a+B
remains const. B

6. Helmholtz free energy of a system in contact with a thermal bath at T
System comes to an equilibrium when the total entropy is maximum
or when its free energy of is minimum.
How much work can we extract from such a system?
[This suggests that we can define an entropic force from free energy]
Maximum possible work:

7. Assuming the initial length is Li, it will expand until
Internal kinetic energy remains the same
The free energy changes by
Introduce
Work done on the load is

8. The difference between free energy and work done is wasted as heat!
In order to extract maximum possible work
Quasi-static processes provide the most efficient way for extracting work
If we bring this system into
contact with a heat bath at
T2 < T1 and recycle this process
we obtain a heat engine. T2 T1

9. Microscopic systems:
Consider a microscopic system in contact
with a heat bath which keeps the temp.
constant at T.
According to the statistical postulate
all the allowed states in the joint system
have the same probability P0
Probability of a particular state in “a”
Boltzmann distribution

10. Two-state system:
Consider a system, which is in thermal equilibrium with a reservoir and
has only two allowed states with energies E1 and E2
The probabilities of being in these states follow from the Boltzmann dist.

11. Activation barrier:
In a more typical situation, there is also an energy barrier in the forward
direction (i.e. E2  E1), which is called the activation barrier, DE‡
Rate constant: (probability of transition per unit time)
For N particles, the rate of flow is given by N.k

12. The forward and backward rate constants and flux are given by
At equilibrium,
Approach to equilibrium
(cf. Nernst pot.)

13. At equilibrium,
To solve the DE, introduce
Here t is called the relaxation (or decay) time to equilibrium.
It critically depends on the activation barrier,
If DE >> kT, k-  0, and transition probability becomes

14. Free energy in microscopic systems:
In a microscopic system, fluctuations in energy can be large, so we need
use the average energy. Given the probability of each state as Pi
Average energy
Entropy (from Shannon’s formula)
By analogy, we define the free energy of a microscopic system as
We need to show that this free energy is minimum at the equilibrium
and leads to the Boltzmann distribution
To find the minimum of Fa, set

15. where Z is the partition function
Substitute in Fa

16. Complex two-state systems:
Because biological macromolecules are flexible, they do not have unique
structures. Rather in a given configuration, there are an ensemble of
states whose energies are very close.
Consider two such configurations I and II, with multiplicities NI and NII
If all the states in I had the same energy EI and in II, EII, then
More generally, they are different, so we need to use a Boltzmann dist.
Using

17. The relative probability becomes
Similarly, the transition rates between two complex states becomes
Note: when the volume changes in a reaction, the quantity to consider is
the Gibbs free energy
This is important for reactions involving the gas phase but can be
ignored in biological systems, where reactions occur in water

18. Example: RNA folding as a two-state system (Bustamante et al, 2001)
Single molecule experiment using optical tweezers
Increasing the force on the bead triggers unfolding of RNA at ~14.5 pN
DNA handle

19. Reversible folding of RNA
Black line: stretching
Gray line: relaxing
Dz = b-a ~ 22 nm
When f = 14.1 pN, half of the RNA
in the sample is folded and half
unfolded
f.Dz = 14.1 x 22 = 310 pN.nm = 76 kT
DF = 79 kT

20. Entropic Forces (Nelson, chap. 7)
Entropic forces are in action when a system in a heat bath does work
e.g., expansion of gas molecules in a box of volume V
Ideal gas law applies equally to solutions

21. Osmotic pressure:
Sugar solution rises in the vessel until it reaches an equilibrium height zf m

22. Osmotic flow
Reverse osmosis mg
van’t Hoff relation

23. Calculate the work done by osmosis when the piston moves from Li  Lf
The sugar concentration changes as
For maximum efficiency, assume that we operate near equilibrium
This is the same expression we calculated for the free energy change
in an expanding volume of gas molecules.
 Dilute solutes in water behave the same way as gas molecules.

24. Osmotic pressure in cells:
Cells are full of solutes; proteins occupy roughly 30% of the cell volume.
Volume of a protein:
Protein concentration:
Note: c(mole/liter)=c(SI)/(103 NA), so c = 0.12 mM
From van’t Hoff relation
(cf. 1 atm = 105 Pa)
Laplace’s formula
for surface tension
Thus S  0.5 x 10-5 x 300 = 1.5 x 10-3 N/m = 1.5 pN/nm, rupture tension!

25. Osmotic pressure and depletion force
When small and large molecules coexist, large ones tend to aggregate.
The short range force driving this process is called the depletion force.
When d < 2R, depletion forces act to reduce the volume inaccessible
to smaller molecules.
d > 2R
d < 2R

26. Electrostatic forces in solutions
Review of electrostatics:
Coulomb’s law:
Electric field:
Gauss’ law:
Differential form:
Central field:
Electric potential:
& energy

27. Dielectrics:
In a dielectric medium charges are screened: q  q/e, r  r/e
where e is the dielectric constant of the medium.
Electric forces, fields and potentials are screened by the same amount.
Water has a large dipole moment and hence a large e (e = 80)
Proteins and lipids have a much lower e (e ≈ 2 - 5)
Boundary between two media with different e forms a polarizable interface
Eex
e1 Induced surface charge density e2

28. e1 = 80 e2 = 2 Examples: 1. Charge near a membrane image charge q d
Estimates of reaction force for an ion near a membrane (q’ ≈ q/84)
Useful quantities to remember
vacuum water
image charge q d
e1 = 80 d q’
e2 = 2

29. e1 = 80 e2 = 2 2. Charge near a protein (spherical) q r
Note that the monopole term (l = 0) vanishes.
The dipole term gives q r
No simple image charge solution
Series solution gives:
e1 = 80
e2 = 2 a

30. e1 = 80 e2 = 2 3. Charge inside a channel (infinite cylinder)
Series solution gives (too complicated!)
The cylinder results are very different because, unlike the other cases,
the charge is inside a closed boundary (cf. induced charge is 40 times
larger). q
e1 = 80
e2 = 2 a

31. Poisson-Boltzmann (PB) equation:
Mobile ions in water respond to changes in the potential so as to
minimize the potential energy of the system.
Instantaneous potential is given by the Poisson eq’n.
At equilibrium, the mobile ions assume a Boltzmann distribution
Substituting this density in the Poisson eq’n gives the PB eq’n
Because of the exp dependence, the PB equation is very difficult to solve.

32. Gouy-Chapman solution of the PB eq’n for a charged surface:
1-1 electrolyte solution (e.g. NaCl)
surface charge density, s
no fixed charges for x > 0
Boundary cond.:
Solution (see the Appendix)
where k is the inverse Debye length, x s

33. Linearized PB equation:
Expand the Boltzmann factor in the ion density as
The first term vanishes from electroneutrality; substituting the second
term in the PB eq’n, we obtain the linearized PB eq’n
This is much easier to solve, e.g. for the previous problem we have
Boundary condition:

34. Charge density:
Debye length controls the thickness of the charge cloud near a charged
surface (at x = 1/k, it drops to about 1/3 of its value at x = 0)
For a 1-1 salt solution at room temperature
Rule of thumb: for a 90 mM solution, 1/k = 10 Å
Although the approximate solution is obtained for
it remains valid up to
For nonlinear effects take over and the potential decays
faster than the exponential.
where c0 is in moles

35. Energy stored in a diffuse layer:
Potential energy due to an external field:
Energy per unit area for a charged surface
Estimate the energy for a unit charge per lipid head group
For a cell of radius 104 nm,
Ion clouds are permanent fixtures of charged surfaces.

36. Debye-Huckel solution of the linearized PB eq’n for a sphere:
Ion clouds modify the Coulomb interaction between macromolecules.
We can solve the PB equation for a sphere to describe this effect.
Represent the total charge q of the molecule with a point charge at r = 0
The coefficients c1 and c2 are determined from the boundary conditions
at r = a, namely, j and eE are continuous across the boundary a q
e=80
e=1

37. Using the second boundary condition gives
Again we have an exponential “Debye” screening of the Coulomb potent.
To find the screening charge use
maximum at r = 1/k

38. Total screening charge
Energy stored in the diffuse layer
For
This is too small, but charges on a protein are on the surface and q >> e
 Coulomb interaction is mediated by the ion clouds around proteins.

39. Why is water special?
Large dipole moment (p = 1.8 Debye)  large dielectric const. (e = 80)
Tetrahedral structure of water
molecules in ice
Molecular structure of H2O
Covalent O-H bond: 1 Å
H-bond distance: 1.7 Å
O-O distance: 2.7 Å
q ~ 0.4 e q q
104o
1 Å
1 Å
-2q

40. Energy scales involved in binding of molecules:
~ 100 kT single covalent bonds, e.g. C-C, C-N, C-O, O-H
~10 kT H-bonds, e.g. O---H, N---H (recall a-helix and DNA)
~1 kT van der Waals attraction between neutral molecules
Note that the above values are for two molecules in vacuum.
In water, as with all other interactions, the H-bond energy is also reduced
to about 1-2 kT. Thus, H-bond energy is comparable to the average K.E.
of 3kT/2, and it can be relatively easily broken.
In ice, there are 4 H-bonds per water molecule.
In water at room temperature, the average number of H-bonds per water
molecule drops to ~3.5. That is, H-bond network is maintained to ~90% !
*** Dynamics of water involves making and breaking of H-bonds ***

41. Solvation of molecules in water:
Ions gain enough energy from solvation so that salt crystals dissolve in
water to separate ions. Born energy for solvation
Energy gain when a monovalent ion goes from vacuum into water:
This energy is larger than the binding energy of an ion in crystal form.
In water, ions have a tightly bound hydration shell around them.
Typically there are 6 water molecules in the first hydration shell.
Solventberg model: conductance of ions in water increases with temp.
for most ions

42. Solvation of nonpolar molecules and hydrophobic interactions:
When nonpolar molecules such as hydrocarbon chains are mixed in
water, they disrupt the H-bond network, costing free energy.
For small molecules, water molecules can form a ‘clathrate cage’ around
the intruder, which almost maintains the H-bond in bulk water.
But such an ordered structure costs entropy, i.e. free energy still goes up.
Size dependence of solubility
of small nonpol. molecules:
Butanol, C4H9OH (top)
Pentanol, C5H11OH
Hexanol, C6H13OH
Heptanol, C7H15OH (bottom)

43. Appendix: Gouy-Chapman solution of PB eq.
Solve the diff. eq.
Subject to the boundary condition:
First introduce the dimensionless potential,
and the inverse Debye length,
The diff. eq. becomes,
Multiply both sides by and integrate from ∞ to x

44. Take the square root of the diff. eq. using
Integrate [0, x]
Substitute
(reject the + solution)

45. Using the identity
We can write
Boundary conditions:

46. Approximate solution for
Boundary condition:
Charge dist.: