Transcript The use of Energy to control cell morphogenesis

The use of Energy to control cell
morphogenesis
Lecture II
Marc W. Kirschner
March 22, 2005
The phenotype depends on the environment. But in the normal
development and function of an organism, the rules are
established by the organism, since so much of the environment
is created by the environment.
But though the environment might be uniform the responses
are only general and not stereotyped.
Endothelial cells migrating
toward the margin of a
wound. The microtubules of
the cytoskeleton are stained.
The arrow is perpendicular
to the wound. From Gotlieb,
et al. ( 1986) J. of Cell
Biology. Note the variety of
structures.
The gene number problem rears
its head again
• There are 22,500 human genes and 1014
neurons, even more synapses
• The definition of systems biology could be
restated in these terms: To discover the
rules by which a small number of genes can
generate the much larger complexity of the
organism
There are three major filament types
in eukaryotic cells:
microtubules
intermediate filaments
actin
Intermediate filaments are different from the
other in that they are not polar, they assemble
without nucleotide hydrolysis, and they have
highly diversified in the recent vertebrate
lineage. They also differ in tensile strength,
which is related to their structure. Microtubules
and actin show unusual properties, involving
nucleotide hydrolysis.
Kinetically linear polymers grow with a two phase growth curve, a nucleation phase and
a linear growth phase. The nucleation phase can be complex.
For sickle cell hemoglobin
the delay can be long and the
polymerization abrupt. The
sickling occurs when the
HbS is deoxygenated in the
capillary. The concentration
dependence of the lag phase
goes as the concentration to
the 80th power!
Simple aggregation theory, with no cooperativity in polymer formation
defines the critical concentration at infinite concentration.
2
K
=
dimer/monomer
=
c
/c
2 1
L
L +L
c2 = Kc12
L
L
L + L
L
K = trimer/dimer•monomer
L
L
K = c /c c
3
c3 = Kc1
2 1
3
In general: ci = K-1(Kc1)i
The total concentration in monomer units is cT
cT = Sici = c1/(1-Kc1)2 {f(x)= Sixi,=x/(1-x)2}
As cT goes to infinity c1 approaches 1/K
Simple linear aggregation of a non-helical polymer
C1 =1/K
c1
10/K
20/K
cT
So if the polymer can form and fall apart a any point the free monomer concentration
grows slowly with the total concentration until it reaches asymptotically c1.
In a helical polymer there is deformation of the subunits to fit into
the polymer. Once a nucleus is formed, however, additional subunits
make bonds with two neighbors and this is more favorable. Thus
there is a factor that makes the initial nucleus unfavorable and a
different factor that makes more favorable subunit addition.
c3h = gc3
And g is a factor <1 which accounts for the additional energy
By the time the fourth subunit adds helically, the K changes
to account for the additional binding surface.
The concentration of a four subunit polymer is:
c4h = Khc3hc1 = gK-1K3Khc14
Where Kh is the binding constant for the binding to the end of the helical polymer.
Kh >>K
The favoring of binding to the end is given by s
s= g(K/Kh)2
It is very small, appx 10-8 for 20 kjoules difference due to g and
20 kjoules to Kh
This leads to a simple equation for the total concentration:
cT = c1 + sc1/(1-Khc1)2
If we look at this equation, at cT< c1; the second term is negligible(s appx 10-8),
so cT is approximately equal to c1 . When cT approaches c1, as c1 approaches 1/Kh,
the second term cannot be ignored. Almost all the subunits go into polymer. The
value of the concentration where this happens (1/Kh) is called the critical
concentration. If this were crystallization, it would be the solubility.
For a helical polymer
c1
ch
cc
cT
From now on we will consider the addition of subunits from the ends only and
ignore nucleation or the first steps. Actin and tubulin, as well as HbS, and
flagellin behave as helical polymers. We will now consider the kinetics of
assembly that holds the clue to the problem we raised initially, what
determines their placement in the cell.
a
aL(solution)
L
L
L
L
L
L
L
L
L
L
L
b
+ end
J
Ja = ac - a
Jb = bc - b
- end
b
L(solution)
-b
-a
ce
By the principle of detailed balance at equilibrium the two ends of
the polymer must have the same critical concentration, which is
another way of saying that despite the differences in rate the
equilibrium constant must be the same for the two ends. Thus
At equilibrium: Ja= Jb = 0 = ac - a = bc - b
Therefore cc = aa = bb = Kh
There is no concentration where the subunits grow from one end
and shrink from the other. If the ends are occluded they will still
have the same equilibrium constant. There is no growth below cc.
In microtubules:
a2
LT
a-2
a-1
a1
LD
LD
LD
LD
LD
LD
LD
Similar processes
At the minus end
In microtubules:
a2
GTP
LT
a-2
GDP LD
LD
LD
LD
LD
LD
LD
a-1
a1
Pi
Similar processes
At the minus end
At the plus (a) end the main
process is the addition of the
tubulin monomer in the GTP
form and the release of the
subunit in the GDP form.
Exchange of GTP for GDP
occurs relatively rapidly in
solution. Therefore the rate
equation on the plus end is:
Ja= a1c - a2
Jb= b1c - b2
As this is steady state and not
equilibrium, it is not true that
both ends have the same critical
concentration.
J
Ja = ac - a
Jb = bc - b
-b
ce
-a
J
cca
ccb
Note the critical
concentrations are not the
same.
J
Ja = ac - a
Jb = bc - b
-b
ce
-a
J
cc
cca
ccb
Note the critical
concentrations are not the
same. At a new steady state
concentration cc, there is a
net flow of subunits on the a
end and a loss of subunits
from the b end. This is
called treadmilling.
This is not the whole story. The coupling of GTP hydrolysis makes it
possible for the off rates of GDP subunits to be very different from that
extrapolated back from the GTP curve. Considering for simplicity the
reactions only at the a end, we have the separate rates for GDP reactions
and GTP reactions:
Scale change
a1c- a-1
J
a1c - a2
a-2c- a2
Adding the reactions at the b end, we have
Scale change
a1c- a-1
a1c - a2
J
a-2c- a2
ce(1)
ce(2)
From Mitchison and Kirschner, 1984
POPULATION BEHAVIOR OF MICROTUBULES