Membrane Transport

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• Membrane Transport

Basic reference Keener and Sneyd, Mathematical
Physiology
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If a picture is worth a thousand words, an
animation is worth a million!
An interesting animated tour of the membrane
transport processes http//www.wiley.com/legacy/
college/boyer/0470003790/animations/membrane_trans
port/membrane_transport.swf
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Basic problem

• The cell is full of stuff. Proteins, ions, fats,
  etc.
• Ordinarily, these would cause huge osmotic
  pressures, sucking water into the cell.
• The cell membrane has no structural strength,
  and the cell would burst.

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Basic solution

• Cells carefully regulate their intracellular
  ionic concentrations, to ensure that no osmotic
  pressures arise
• As a consequence, the major ions Na, K, Cl-
  and Ca2 have different concentrations in the
  extracellular and intracellular environments.
• And thus a voltage difference arises across the
  cell membrane.
• Essentially two different kinds of cells
  excitable and nonexcitable.
• All cells have a resting membrane potential, but
  only excitable cells modulate it actively.

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Typical ionic concentrations (in mM)
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The cell at steady state
Na,K-ATPase

• We need to model
• pumps and exchangers
• ionic currents
• osmotic forces

Calcium ATPase
Ill talk about this a lot more in week 6.
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Active pumping

• Clearly, the action of the pumps is crucial for
  the maintenance of ionic concentration
  differences
• Many different kinds of pumps. Some use ATP as
  an energy source to pump against a gradient,
  others use a gradient of one ion to pump another
  ion against its gradient.
• A huge proportion of all the energy intake of a
  human is devoted to the operation of the ionic
  pumps.
• Not all that many pump models that I know of. It
  doesn't seem to be a popular modelling area. I
  have no idea why.

http//www.northland.cc.mn.us/biology/BIOLOGY1111/
animations/active1.swf
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A Simple ATPase
flux
Note how the flux is driven by how far the
concentrations are away from equilibrium
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Reducing this simple model
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Na-K ATPase
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Na-K ATPase (Post-Albers)
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Simplified Na-K ATPase
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The cell at steady state

• We need to model
• pumps
• ionic currents
• osmotic forces

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The Nernst equation
(The Nernst potential)
Note equilibrium only. Tells us nothing about
the current. In addition, there is very little
actual ion transfer from side to side. We'll
discuss the multi-ion case later.
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Only very little ion transfer
spherical cell - radius 25 mm surface area - 8 x
10-5 cm2 total capacitance - 8 x 10-5 mF
(membrance capacitance is about 1 mF/cm2) If the
potential difference is -70 mV, this gives a
total excess charge on the cell membrane of about
5 x 10-12 C. Since Faraday's constant, F, is
9.649 x 104 C/mole, this charge is equivalent to
about 5 x 10-15 moles. But, the cell volume is
about 65 x 10-9 litres, which, with an internal
K concentration of 100 mM, gives about 6.5 x
10-9 moles of K. So, the excess charge
corresponds to about 1 millionth of the
background K concentration.
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Electrical circuit model of cell membrane
How to model this is the crucial question
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How to model Iionic

• Many different possible models of Iionic
• Constant field assumption gives the
  Goldman-Hodgkin-Katz model
• The Poisson-Nernst-Planck (PNP) equations can
  derive expressions from first principles
  (Eisenberg and others)
• Barrier models, binding models, saturating
  models, etc etc.
• Hodgkin and Huxley in their famous paper used a
  simple linear model
• Ultimately, the best choice of model is
  determined by experimental measurements of the
  I-V curve.

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Two common current models
Linear model
GHK model
These are the two most common current models.
Note how they both have the same reversal
potential, as they must. (Crucial fact In
electrically excitable cells gNa (or PNa) are not
constant, but are functions of voltage and time.
More on this later.)
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Electrodiffusion deriving current models
Poisson-Nernst-Planck equations. PNP equations.
Poisson equation and electrodiffusion
Boundary conditions
Nernst-Planck equations.
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The short-channel limit
If the channel is short, then L 0 and so l 0.
This is the Goldman-Hodgkin-Katz equation. Note
a short channel implies independence of ion
movement through the channel.
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The long-channel limit
If the channel is long, then 1/L 0 and so 1/l
0.
This is the linear I-V curve. The independence
principle is not satisfied, so no independent
movement of ions through the channel. Not
surprising in a long channel.
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The cell at steady state

• We need to model
• pumps and exchangers
• ionic currents
• osmotic forces

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Osmosis
P1
P2
water Solvent (conc. c)
water
At equilibrium
Note equilibrium only. No information about the
flow.
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A Model of Volume Control
Putting together the three components (pumps,
currents and osmosis) gives.....
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The Pump-Leak Model
cell volume
Nai
pump rate
Note how this is a really crappy pump model
Na is pumped out. K is pumped in. So cells have
low Na and high K inside. For now we ignore
Ca2. Cl- just equilibrates passively.
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Charge and osmotic balance

• The proteins (X) are negatively charged, with
  valence zx.
• Both inside and outside are electrically
  neutral.
• The same number of ions on each side.
• 5 equations, 5 unknowns (internal ionic
  concentrations, voltage, and volume). Just solve.

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Steady-state solution
If the pump stops, the cell bursts, as
expected. The minimal volume gives approximately
the correct membrane potential. In a more
complicated model, one would have to consider
time dependence also. And the real story is far
more complicated.
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Ion transport

• How can epithelial cells transport ions (and
  water) while maintaining a constant cell volume?
• Spatial separation of the leaks and the pumps is
  one option.
• But intricate control mechanisms are needed
  also.
• A fertile field for modelling. (Eg. A.Weinstein,
  Bull. Math. Biol. 54, 537, 1992.)

The KJU model. Koefoed-Johnsen and Ussing (1958).
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Steady state equations
Note the different current and pump models
electroneutrality
osmotic balance
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Transport control
Simple manipulations show that a solution exists
if
Clearly, in order to handle the greatest range of
mucosal to serosal concentrations, one would want
to have the Na permeability a decreasing
function of the mucosal concentration, and the K
permeability an increasing function of the
mucosal Na concentration. As it happens, cells
do both these things. For instance, as the cell
swells (due to higher internal Na
concentration), stretch-activated K channels
open, thus increasing the K conductance.
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Lots of interesting unsolved problems

• How do organsims adjust to dramatic environmental
  changes (T. Californicus)?
• How do plants (especially in arid regions)
  prevent dehydration in high salt environments?
  (They make proline.)
• How do plants breathe?
• How do fish (salmon) deal with both fresh and
  salt water?
• What happens to a cell and its environment when
  there is ischemia?

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Stomata control the uptake of carbon dioxide
(photosynthesis) and the loss of water vapour
CO2 uptake
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• Plants use and lose a lot of water
• a one hectare wheat crop will lose 60t of
  water a day ( 8 mm of rain)
• an average sized oak tree will lose 120kg of
  water a day
• Evaporative water loss is
• controlled by pores on
• the leaf surface called stomata

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Stomata are pores in the surfaces of leaves.
Stomata are bounded by two guard cells. The
aperture of the stomatal pore is controlled by
the two guard cells.
What do they do? Stomata control the exchange
of gases between the interior of the leaf and the
atmosphere.
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Why are stomata they important?
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High turgor pressure Low turgor pressure
Opening is associated with water entering the
guard cells. This causes them to swell.
Thickenings in the cell wall cause the guard
cells to bow open causing the pore to open.
Conversely a loss of water causes the cells to
shrink and the pore closes.
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CLOSED
OPEN Guard cells integrate information from
environmental signals to set the most
appropriate stomatal aperture to suit the
prevailing conditions.
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Plant cells are surrounded by a cell wall. This
restricts the expansion of the cell. The cell
wall is made of (among other things) cellulose.
In the case of the guard cell the cellulaose
fibrils are arranged radially.