Molecular machines in membranes

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Chapter 11 Biological Physics Nelson Updated 1 st Edition Slide 1-1 Molecular machines in membranes

Transcript of Molecular machines in membranes

Page 1: Molecular machines in membranes

Chapter 11

Biological PhysicsNelson

Updated 1st Edition

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Molecular machines in membranes

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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)

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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.

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What is all the Na+ and K+ about?

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Electro-osmotic effects

11.1

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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

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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

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4.6.3 Nernst relation review

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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

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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

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11.1.2 A giant axon (one can literally insert

an electrode into these axons)

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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 ...

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Fig

ure

11

.2 (

Ske

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gra

ph

s.)

Cap

tio

n: S

ee

te

xt.

Permeant K+

C D

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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).

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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

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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

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Your Turn 11 A

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• 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?

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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- .

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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)

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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

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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?

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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 …

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ION PUMPING

11.2

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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

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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

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Analogy of non equilibrium steady state

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• 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

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Circuit model for path of cell membrane

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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.

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• 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

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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)

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Modified model for path of cell membrane

Vpump

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Radioactive labeled Na+ experiment

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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).

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Skou’s experiment

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• 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?

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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:

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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)

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Mitochondria as factories

11.3

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11.3.1 Busbars and driveshafts distribute

energy in factories (factory circa. 1820)

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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

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Fig. 11.8 Mitochondrian factory

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Your Turn 11D (Krebs cycle)Confirm Krebs cycle is properly balanced. Hint use Fig. 2.12.

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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.

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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

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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?

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Your Turn 11E Solution

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Slide 1-49©1993. Used by permission of Springer-Verlag.

Fig. 11.9 Oxidative phosphorylation

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A more realistic image

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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

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ATP Synthase

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11.3.5 Vista: Cells use chemiosmotic

coupling in many other contexts

• Discussion/extra reading

• Proton pumps

• Other pumps (ATPase)

• The flagellar motor

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Excursion: Powering up the

flagellar motor

11.4

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Slide 1-55Reprinted with permission from Nature. ©1995, Macmillan Magazines Ltd.

11.4 Extra reading

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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)