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Chapter 11
Biological PhysicsNelson
Updated 1st Edition
Slide 1-1
Molecular machines in membranes
Slide 1-2
Contents
• 11.1 Electro-osmotic effects
– Nernst-Planck formula
• 11.2 Ion pumping
– K+ and Na+ ion pumping and the membrane
potential
• 11.3 Mitochondria as factories
– 11.3.2-11.3.3 Using protons rather than ions
– 11.3.4 and 11.3.5 (extra reading/discussion)
• 11.4 Excursion: “Powering up the flagella motor”
(extra reading)
Slide 1-3
The BIG question
• Biological question: The cytosol’s composition is
very different from that of the outside world. Why
doesn’t osmotic flow through the plasma
membrane burst (or shrink) the cell?
• Physical idea: Active ion pumping by molecular
machines can maintain a nonequilibrium
osmotically regulated (steady) state.
Slide 1-4
What is all the Na+ and K+ about?
Slide 1-5
Electro-osmotic effects
11.1
Slide 1-6
11.1.1 Before the ancients
• The separation of the sciences is a recent thing.
• Previously we had people like Franklin & Volta
who were using biological systems to
demonstrate the newly discovered “Electricity”
– Franklin:- Kite experiment in storms
– Galvani:- Frogs legs twitch when hung on two
different metal hooks
• Galvani believed electricity was generated by
the frogs muscles; Volta thought the opposite:-
An electric signal is carried by muscles
– Hey went on to invent Voltac cell to prove this
Slide 1-7
11.1.1 ...
• Volta was a little too harsh to dismiss Galvani’s
idea and in the 1850s Redmond showed a
potential difference of 100 mv between either
side of frog skin
• The concept of the cell membrane as an
electrical insulator only a few nanometers thick
remained speculation until 1927, when Fricke
measured the capacitance of a cell membrane
and thus estimated its thickness ...
• Now let’s quickly review the Nernst Relation
Slide 1-8
4.6.3 Nernst relation review
Slide 1-9
4.6.3 Nernst relation & scale of cell
membrane potentials
• Consider now a charged situation like many cell
membranes in biology (see Fig. 4.14)
• The electric field E = ΔV/l and hence the drift
velocity is
• Now consider a flux trough area A (Fig. 4.14)
and we argue that j = c vdrift which implies that
• Now including dissipation in Fick’s law we find
and using the Einstein relation we find
Slide 1-10
The Nernst-Planck Formula
• FQ:- what electric field will cancel out non-
uniformity in a solution?
• Ans:- Set j=0 implies which has
solution
where ΔV = EΔx
• Using real values we estimate ΔV~58 mV
Slide 1-11
11.1.2 A giant axon (one can literally insert
an electrode into these axons)
Slide 1-12
11.1.2 Ion concentration differences cause
Nernst potentials
• What do the results of Section 4.6.3 have
anything to do with cells?
• Consider Figure 11.1 & two concentrations c2>c1
(far from the membrane) for each compartment
(assume monovalent K+Cl- solution)
• The membrane is slightly permeable to K+ but
not to Cl- (ignoring osmotic flow of water)
• After system equilibrates c+ and c- will not be
uniform (see Fig. 11.2a)
• We need to now explain the potential in Fig.
11.2b? Read relevant pages ...
Slide 1-13
Fig
ure
11
.2 (
Ske
tch
gra
ph
s.)
Cap
tio
n: S
ee
te
xt.
Permeant K+
C D
Slide 1-14
Figure 11.2 Caption
(a) Concentration profiles near a membrane, for the situation
sketched in Figure 11.1. Far outside the membrane (r→∞) the
concentrations c± of positive and negative ions must be equal, by
charge neutrality; their common value c1 is just the exterior salt
concentration. Similarly, deep inside the cell c+ = c− = c2. The
situation shown assumes that only the positive ions are permeant.
Thus some positive ions leak out, enhancing c+ in a layer of
thickness λ just outside the membrane and depleting it just inside.
c− drops just outside the membrane, because negative ions move
away from the negatively charged cell. The concentrations in the
membrane’s hydrophobic interior (the region between B and C)
are nearly zero. (b) The corresponding electrical potential V
created by the charge distribution in (a). In equilibrium, ΔV
equals the Nernst potential of the permeant species (in this case
the positive ions).
Slide 1-15
Qualititive explanation of 11.2b
• K+ could increase ΔS by crossing membrane to
erase Δc, but Cl- will always call them back by
attraction (assuming Cl- is constant)
• Leads to depletion of K+ just inside the mem-
brane and restoration in thin layer just outside
• Point C is salt solution in contact with a
– surface (the membrane) and therefore
also leads to a + cloud (see right)
• This layer is region CD and is enhanced
-depleted by K+ /Cl-
• Region AB is juxtaposed: salt solution facing
a + membrane surface
Slide 1-16
continued
• Now bring in a + test charge (TC) from r→∞. At
first the membrane has zero net charge so no
change in potential
• However once TC gets to D it is attracted to –
membrane wall at C → its potential decreases
• The deeper into the cloud the deeper the drop in
potential
• Remember the membrane is meant to be neutral
so it feels a constant force attracting it to inner
membrane wall B, implying a linear falling
potential in BC
Slide 1-17
Your Turn 11 A
Slide 1-18
• What happens to Fig 11.2 now if still c2>c1 but the
membrane is permeable to Cl- not K+? What
changes - Draw a picture & show me next time!
• How do we determine drop ΔV?
– Using the Nernst relation and assuming a
battery across the cell in reverse to just stop
the current then in equilibrium
where
• In equilibrium, electrochemical potential of any
permeant ion species must be same everywhere.
If not implies cells are non-equilibrium!
Questions?
Slide 1-19
11.1.3 Donnan equilibrium can create a resting
membrane potential
• The Nernst relation gives the potential arising
when a permeant species reaches equilibrium.
Equivalently, it gives the potential that must be
applied to stop the net flux of that species, given
the concentration jump across a membrane
• Now consider more than two ion species (3 for
simplicity): Na+, K+, Cl- with concentrations ci,
i=1,2,3
• Note, the cell can import some more Na+while
remaining neutral, if at the same time it expels
some K+ or pulls in some Cl- .
Slide 1-20
11.1.3 Typical values of charge densities?
• The book states the following charge for trapped
impermeants ρq,macro ~ 125 mM and assumes
cell sits in infinite bath of concentration c1,i.
• Examples are: c1,Na+ = 140 mM, c1,K+ = 10 mM
and c1,Cl- = 150 mM which is neutral:
• The cell’s interior is not infinite so concentrations
c2,i are not fixed and we also need a fourth
unknown quantity, ΔV=V2 –V1. Neutrality implies
from Gauss’s law (we have a charge membrane)
Slide 1-21
11.3.1 Gibbs-Donnan relations
• Same electrostatic potential function affects
every ion species. And in equilibrium each
species must be in Nernst equilibrium:
• To solve this and interior charge neutrality
relation (previous slide) note the above can be
written as the Gibbs-Donnan relations:
• Now try example on pg. 475
Slide 1-22
Example on page 475
a. Why is the Cl ratio in the Gibbs-Donnan relation
inverted relative to the others?
b. Finish calculation & find c2,i and ΔV?
Slide 1-23
What does this mean
• It means that Donnan equilibrium gives the
wrong estimate for the sign of the Sodium
(Nernst) equilibrium potential
• This is explained in the next section …
Slide 1-24
ION PUMPING
11.2
Slide 1-25
11.2.1 Eukaryotic cells far from
Donnan equilibrium
• Donnan equilibrium looks a promising way to
model the cell membrane potential but we also
not yet considered osmotic pressure effects:
• This means to stop inward osmotic flow the
membrane would need to maintain an interior
pressure of 25 mM× kBTr ~ 6・104 Pa, but
Eukaryotic cells burst at much lower pressures!
• Bilayer cells (plants, bacteria etc.) can manage
this. What about animal cells?
– See Table 11.1
Slide 1-26
11.2.1 The sodium anomaly (Table 11.1)
• The Nernst potential of sodium is much more positive
than the actual membrane potentials ΔV
• All animal cells (not just squid axons) suffer from
this kind of sodium anamoly; plants suffer from a
proton anamoly (see section 11.3)
• Ions in a cell are not in equilibrium (pg 477-478),
but can give a constant flux (steady state) of ions
Slide 1-27
Analogy of non equilibrium steady state
Slide 1-28
• Based on Nernst-Planck formula (Chapter 4), if
out of equilibrium then we expect an overall flux
of ion species i, Eq. (11.8):
• Compare to Chapter 7:
for osmotic forces. Equilibrium → Δp=ΔckBT
• This is just another form of Ohm’s law: Electric
current I through a patch of membrane is jqA and
then ΔV=IR+ 𝓥 Nernst, where R = 1/(gA), see Fig.
11.4. Note above Eqn. is only an approximation
(read pg. 481)
11.2.2 Ohmic conductance hypothesis
Slide 1-29
Circuit model for path of cell membrane
Slide 1-30
Your Turn 11C
• Find the relation between the conductance per
area and the permeability of a membrane to a
particular ion species, assuming that the inside
and outside concentrations are nearly equal.
Slide 1-31
• The sodium anomaly was measured around
1948 using radioactive Na+ on one side of the
membrane and measured leakage across
boundary: implying non-
equilibrium
• Further experiments on Frog skin with Δ𝑉 = 0still lead to Na pumping implied modification to
Eq. (11.8):
• Further experiments (pg 482) lead to the idea:-
11.2.3 Active pumping maintains steady-state
(non equilibrium) membrane potentials avoiding
large osmotic pressures
Slide 1-32
11.2.3 Idea 11.11:-
• A specific molecular machine embedded in cell
membranes hydrolyzes ATP, then uses some of
the resulting free energy to pump sodium ions
out of the cell. At the same time the pump
imports potassium, partially offsetting the loss of
electric charge from the exported sodium.
– See Figure 11.5
• The pump operates only when sodium and ATP
are available on its inner side and potassium is
available on its outer side. If any of these are
stopped, the cell slowly reverts to concentrations
appropriate for equilibrium (Fig. 11.6)
Slide 1-33
Modified model for path of cell membrane
Vpump
Slide 1-34
Radioactive labeled Na+ experiment
Slide 1-35
11.2.3 What was doing the pumping
• Prior to 1955 no specific membrane constituent
was even known to be a candidate for ion
pumping
– From the possible thousands of
transmembrane proteins available
• Then in 1957, Skou the isolated from crab
membranes a single membrane protein with
ATPase activity (see Fig. 11.6)
• Interestingly, the enzyme required both Na and
K, which is the same behaviour Hodgkin &
Keynes found for whole squid axons (Fig. 11.5).
Slide 1-36
Skou’s experiment
Slide 1-37
• Compare the free energy gain from hydrolyzing
one ATP molecule to the cost of running the
pump through a cycle
• Using Table 11.1 and total free-energy cost to
pump one sodium ion out is
potassium ions in is
which are both positive and energy cost/cycle is
(3e×114 mV)+(2e×15 mV)=0.372 eV=15kBT
• But ATP hydrolysis liberates ~19kBT and so only
4kBT lost is lost as heat (pretty efficient, yes!)
Is such a pump energetically favorable?
Slide 1-38
Ion pumping mechanism (cf. Table 11.1)
• There must be no net flux of any ion; otherwise
some ion would pile up somewhere, eventually
changing the concentrations. Every ion must
either be impermeant (like the interior macro-
molecules), or in Nernst equilibrium, or actively
pumped.
• Model assumes jpumpK+=-2/3jpump
Na+ with
definition jpumpNa+>0. Steady state implies
jK+=jNa+=0 or
• Cl is permeant and not pumped and ΔV=-60 mV
is in good agreement. Na and K however imply
Ohmic part of fluxes must be in ratio of -2/3:
Slide 1-39
Ion pumping mechanism (continued)
• Solving for ΔV then gives
The ion species with the greatest conductance/area
gets the biggest vote in determining the steady-state
membrane potential. That is, the resting membrane
potential ΔV is closer to the Nernst potential of the
most permeant pumped species (here VNernstK+), than
it is to that of the less permeant ones (here VNernstNa+).
• If membrane can switch to conducting Na+ better than
K+ then membrane potential reverses sign! (Chap 12)
Slide 1-40
Mitochondria as factories
11.3
Slide 1-41
11.3.1 Busbars and driveshafts distribute
energy in factories (factory circa. 1820)
Slide 1-42
11.3.2 The biochemical backdrop to
respiration
• Oxidation & reduction. Anaerobic cells only
create two ATPs with pyruvate as waste (to
lactate etc.)
• Aerobic case was a mystery until 1948 Kennedy
and Lehninger found mitochondrion site
• Fig. 11.8 shows:-
– Decarboxylation of pyruvate
– Krebs (tricarboxylic acid) cycle
Slide 1-43
Fig. 11.8 Mitochondrian factory
Slide 1-44
Your Turn 11D (Krebs cycle)Confirm Krebs cycle is properly balanced. Hint use Fig. 2.12.
Slide 1-45
11.3.2 Summarized
Reactions equations 11.14 and 11.15 oxidize pyruvate
completely: Pyruvate’s three carbon atoms each end up
as molecules of carbon dioxide. Conversely, four
molecules of the carrier NAD+ and one of FAD get
reduced to NADH and FADH2. Since glycolysis
generates two molecules of pyruvate and two of NADH,
the overall effect is to generate ten NADH and two
FADH2 per glucose. Two ATP per glucose have been
formed so far from glycolysis, and the equivalent of two
more from the citric acid cycle.
Slide 1-46
11.3.3 Chemiosmotic mechanism identifies
mitochondrial inner membrane as busbar
• Peter Mitchell 1961: Generation, Transmission &
Utilization (just like at a power plant)
• Generation: The final oxidation reaction in a
mitochondrion (respiration) is (similar for FADH2)
• Transmission: Inner membrane is impermeable
to protons so once pumped out, electrochemical
potential difference spreads over whole surface
• Utilization: ATP synthase allows protons back
into membrane but coupled to ATP synthesis
• 10 NADH & 2 FADH → 10×2.5+2×1.5=28 ATPs
Slide 1-47
Your Turn 11E
a. Adapt the logic of Example 1 to find the difference in
electrochemical potential for protons across the mitochondrial
inner membrane. Use the following experimental input: The pH
in the matrix minus that outside is ΔpH=1.4, while the
corresponding electrical potential difference equals ΔV ≈ −0.16
volt.
b. The difference you just found is often expressed as a “proton-
motive force,” or “p.m.f.,” defined as (ΔμH+)/e. Compute it,
expressing your answer in volts.
c. Compute the total ΔG0NAD + 10ΔμH+ for the coupled oxidation
of one molecule of NADH and transport of ten protons. Is it
reasonable to expect this reaction to go forward? What
information would you need in order to be sure?
Slide 1-48
Your Turn 11E Solution
Slide 1-49©1993. Used by permission of Springer-Verlag.
Fig. 11.9 Oxidative phosphorylation
Slide 1-50©1993. Used by permission of Springer-Verlag.
A more realistic image
Slide 1-51
11.3.4 Evidence for chemiosmotic
mechanism?
• Discussion/Class reading
• Independence of generation and utilization
• Membrane as an electrical insulation
• Operation of the ATP synthase
Slide 1-52©2000. Used by permission of Elsevier Science.
ATP Synthase
Slide 1-53
11.3.5 Vista: Cells use chemiosmotic
coupling in many other contexts
• Discussion/extra reading
• Proton pumps
• Other pumps (ATPase)
• The flagellar motor
Slide 1-54
Excursion: Powering up the
flagellar motor
11.4
Slide 1-55Reprinted with permission from Nature. ©1995, Macmillan Magazines Ltd.
11.4 Extra reading
Slide 1-56
The Big Picture
• Before the enzyme for ion pumps was
discovered the physics was already indicating
their existence (much like DNA)
• Osmotic pressure would destroy a eukaroytic
cell, but using the sodium anomaly and moving
to non-equilibrium we can solve these two
problems using steady state ion pumping
• Chapter 12 will see how voltage gated ion
channels turn a charged membrane into an
exctable medium (nerve axons)