V22 Modelling Dynamic Cellular Processes
-
Upload
ferdinand-leonard -
Category
Documents
-
view
34 -
download
2
description
Transcript of V22 Modelling Dynamic Cellular Processes
22. Lecture WS 2005/06
Bioinformatics III 1
V22 Modelling Dynamic Cellular Processes
John Tyson Bela Novak
Mathematical description of signalling
pathways helps answering questions like:
(1) How do the magnitudes of signal output
and signal duration depend on the kinetic
properties of pathway components?
(2) Can high signal amplification be coupled
with fast signaling?
(3) How are signaling pathways designed to
ensure that they are safely off in the absence
of stimulation, yet display high signal
amplification following receptor activation?
(4) How can different agonists stimulate the
same pathway in distinct ways to elicit a
sustained or a transient response, which can
have dramatically different consequences?
Heinrich et al. Mol. Cell. 9, 957 (2002)
22. Lecture WS 2005/06
Bioinformatics III 2
The Cyclin – E2F cell cycle control system
as
Kohn, Molec. Biol. Cell 1999 10:2703-34
22. Lecture WS 2005/06
Bioinformatics III 3
Biocarta pathways
http://cgap.nci.nih.gov/Pathways/BioCarta/h_mapkPathway
22. Lecture WS 2005/06
Bioinformatics III 4
Ras signaling pathway
http://cgap.nci.nih.gov/Pathways/BioCarta/h_mapkPathway
22. Lecture WS 2005/06
Bioinformatics III 5
Biocarta pathwaysIncredible amount
of information about
signalling pathways!
Constantly updated.
How can we digest this?
Need computational
models!
22. Lecture WS 2005/06
Bioinformatics III 6
Protein synthesis and degradation: linear response
RkSkkdt
dR210
S = signal strength (e.g. concentration of mRNA)
R = response magnitude (e.g. concentration of protein)
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
A steady-state solution of a differential equation, dR/dt = f(R) is a constant Rss
that satisfies the algebraic equation f(Rss) = 0. In this case
2
10
k
SkkRss
22. Lecture WS 2005/06
Bioinformatics III 7
Protein synthesis and degradation
RkSkkdt
dR210
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
Solid curve: rate of removal of the
response component R.
Dashed lines: rates of production of R
for various values of signal strength.
Filled circles: steady-state solutions
where rate of production and rate of
removal are identical.
Wiring diagram rate curve signal-response curve
Steady-state response R as a
function of signal strength S.
2
10
k
SkkRss
Parameters:k0 = 0.01k1 = 1k2 = 5
22. Lecture WS 2005/06
Bioinformatics III 8
phosphorylation/dephosphorylation: hyperbolic response
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
TP
PPTP
PPPT
SRkSkkR
SRkSRkSRkRk
RkRkRRSk
112
1112
2210
A steady-state solution:
Skk
SRR T
ssP
12
,
PPT
P RkRRSkdt
dR21
RP: phosphorylated form of the response element
RP= [RP]
Pi: inorganic phosphate
RT = R + RP total concentration of R
Parameters:k1 = k2 = 1RT = 1
22. Lecture WS 2005/06
Bioinformatics III 9
phosphorylation/dephosphorylation: sigmoidal response
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
Modification of (b) where the phosphorylation and
dephosphorylation are governed by Michaelis-
Menten kinetics:
Pm
P
PTm
PTP
RK
Rk
RRK
RRSk
dt
dR
2
2
1
1
PmPTPTmP
PTm
PT
Pm
P
Pm
P
PTm
PT
RKRRSkRRKRk
RRK
RRSk
RK
Rk
RK
Rk
RRK
RRSk
2112
1
1
2
2
2
2
1
10
steady-state solution of RP The biophysically acceptable solutions
must be in the range 0 < RP < RT:
T
m
T
m
T
ssP
R
K
R
KkSkG
R
R21
21
, ,,,
with the „Goldbeter-Koshland“ function G:
uKuvuKvJuvuKvJuv
uK
KJvuG
4
2
),,,(
2
22. Lecture WS 2005/06
Bioinformatics III 10
phosphorylation/dephosphorylation: sigmoidal response
This mechanism creates a switch-like signal-response curve which is
called zero-order ultrasensitivity.
(a), (b), and (c) give „graded“ and reversible behavior of R and S.
„graded“: R increases continuously with S
reversible: if S is change from Sinitial to Sfinal , the response at Sfinal is the same
whether the signal is being increased (Sinitial < Sfinal) or decreased (Sinitial > Sfinal).
Element behaves like a buzzer: to activate the response, one must push hard
enough on the button.Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
Parameters:k1 = k2 = 1RT = 1Km1 = Km2 = 0.05
22. Lecture WS 2005/06
Bioinformatics III 11
perfect adaptation: sniffer
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
Here, the simple linear response element is supplemented
with a second signaling pathway through species X.
XkSkdt
dX
RXkSkdt
dR
43
21
32
41
2
1
4
3
kk
kk
Xk
SkR
k
SkX
ss
ss
The response mechanism exhibits perfect adaptation to the signal:
although the signaling pathway exhibits a transient response to changes in signal
strength, its steady-state response Rss is independent of S.
Such behavior is typical of chemotactic systems, which respond to an abrupt
change in attractants or repellents, but then adapt to a constant level of the signal.
Our own sense of smell operates this way we call this element „sniffer“.
22. Lecture WS 2005/06
Bioinformatics III 12
perfect adaptation
Right panel: transient response (R, black) as a function of stepwise increases
in signal strength S (red) with concomitant changes in the indirect signaling
pathway X (green).
The signal influences the response via two parallel pathways that push the
response in opposite directions. This is an example of feed-forward control.
Alternatively, some component of a response pathway may feed back on the
signal (positive, negative, or mixed).
Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003)
Parameters:k1 = k2 = 2k3 = k4 = 1
22. Lecture WS 2005/06
Bioinformatics III 13
Positive feedback: Mutual activation
Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003)
E: a protein involved with R
EP: phosphorylated form of E
Here, R activates E by phosphorylation,
and EP enhances the synthesis of R.
4343
210
,,, JJkRkGRE
RkSkREkdt
dR
P
P
22. Lecture WS 2005/06
Bioinformatics III 14
mutual activation: one-way switch
As S increases, the response is low until S exceeds a critical value Scrit at which
point the response increases abruptly to a high value.
Then, if S decreases, the response stays high.
between 0 and Scrit, the control system is „bistable“ – it has two stable
steady-state response values (on the upper and lower branches, the solid lines)
separated by an unstable steady state (on the intermediate branch, the dashed
line).
This is called a one-parameter bifurcation. Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003)
22. Lecture WS 2005/06
Bioinformatics III 15
mutual inhibition
Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003)
Here, R inhibits E, and E promotes the degradation of R.
4343
'
2210
,,, JJRkkGRE
RREkRkSkkdt
dR
22. Lecture WS 2005/06
Bioinformatics III 16
mutual inhibition: toggle switch
This bifurcation is called toggle switch („Kippschalter“):
if S is decreased enough, the switch will go back to the off-state.
For intermediate stimulus strengh (Scrit1 < S < Scrit2), the response of the system
can be either small or large, depending on how S was changed.
This is often called „hysteresis“.
Examples: lac operon in bacteria, activation of M-phase promoting factor in frog
egg extracts, and the autocatalytic conversion of normal prion protein to its
pathogenic form.Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003)
22. Lecture WS 2005/06
Bioinformatics III 17
Negative feedback: homeostasis
Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003)
In negative feedback, the response counteracts the effect of the stimulus.
Here, the response element R inhibits the enzyme E catalyzing its synthesis.
Therefore, the rate of production of R is a sigmoidal decreasing function of R.
4343
20
,,, JJRkkGRE
RSkREkdt
dR
Negative feedback in a two-component
system X R | X can also exhibit
damped oscillations to a stable steady state
but not sustained oscillations.
22. Lecture WS 2005/06
Bioinformatics III 18
Negative feedback: oscillatory response
Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003)
There are two ways to close the negative feedback loop:
(1) RP inhibits the synthesis of X
(2) RP activates the degradation of X.
Sustained oscillations require at least 3 components:
X Y R |X
Left: example for a negative-feedback control loop.
Pm
P
PTm
PTPP
Pm
P
PTm
PTP
P
RK
Rk
RRK
RRYk
dt
dR
YK
Yk
YYK
YYXk
dt
dY
XRkXkSkkdt
dX
6
6
5
5
4
4
3
3
'
2210
22. Lecture WS 2005/06
Bioinformatics III 19
Negative feedback: oscillatory response
Feedback loop leads to
oscillations of X (black),
YP (red), and RP (blue).
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
Within the range Scrit1 < S
< Scrit2, the steady-state
response RP,ss is unstable.
Within this range, RP(t)
oscillates between RPmin
and RPmax.
Again, Scrit1 and Scrit2 are bifurcation points.
The oscillations arise by a generic mechanism
called „Hopf bifurcation“.
Negative feedback has ben proposed as a basis for
oscillations in protein synthesis, MPF activity, MAPK
signaling pathways, and circadian rhythms.
22. Lecture WS 2005/06
Bioinformatics III 20
Positive and negative feedback: Activator-inhibitor oscillations
R is created in an autocatalytic
process, and then promotes the
production of an inhibitor X,
which speeds up R removal.
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
4343
65
'
2210
,,, JJkRkGRE
XkRkdt
dX
RXkRkSkREkdt
dR
P
P
The classic example of such a system is cyclic AMP production in the slime mold. External cAMP binds to a surface receptor, which stimulates adenylate cyclase to produce and excrete more cAMP. At the same time, cAMP-bindingpushes the receptor into an inactive form. After cAMP falls off, the inactive form slowly recovers its ability tobind cAMP and stimulate adenylate cyclase again.
22. Lecture WS 2005/06
Bioinformatics III 21
Substrate-depletion oscillations
X is converted into R in an autocatalytic process.
Suppose at first, X is abundant and R is scarce.
As R builds up, the production of R accelerates until there is an explosive
conversion of the entire pool of X into R. Then, the autocatalytic reaction shuts
off for lack of substrate, X. R is degraded, and X must build up again.
This is the mechanism of MPF oscillations in frog egg extract.
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
22. Lecture WS 2005/06
Bioinformatics III 22
Complex networks
All the signal-response elements just described, buzzers, sniffers, toggles and
blinkers, usually appear as components of more complex networks.
Example: wiring diagram for the Cdk network regulating DNA synthesis and mitosis.
The network involving proteins that regulate the activity of Cdk1-cyclin B
heterodimers consists of 3 modules that oversee the
- G1/S
- G2/M, and
- M/G1 transitions of the cell cycle.
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
22. Lecture WS 2005/06
Bioinformatics III 23
Cell cycle control system
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
22. Lecture WS 2005/06
Bioinformatics III 24
Cell cycle control system
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
The G1/S module is a toggle switch, based on mutual inhibition between
Cdk1-cyclin B and CKI, a stoichiometric cyclin-dependent kinase inhibitor.
22. Lecture WS 2005/06
Bioinformatics III 25
Cell cycle control system
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
The G2/M module is a second toggle switch, based on mutual activation between
Cdk1-cyclinB and Cdc25 (a phosphotase that activates the dimer) and mutual
inhibition between Cdk1-cyclin B and Wee1 (a kinase that inactivates the dimer).
22. Lecture WS 2005/06
Bioinformatics III 26
Cell cycle control system
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
The M/G1 module is an oscillator, based on a negative-feedback loop:
Cdk1-cyclin B activates the anaphase-promoting complex (APC), which
activates Cdc20, which degrades cyclin B.
The „signal“ that drives cell proliferation is cell growth: a newborn cell cannot
leave G1 and enter the DNA synthesis/division process (S/G2/M) until it grows
to a critical size.
22. Lecture WS 2005/06
Bioinformatics III 27
Cell cycle control system
The signal-response curve is
a plot of steady-state activity
of Cdk1-cyclin B as a
function of cell size.
Progress through the cell cycle is
viewed as a sequence of bifurcations.
A very small newborn cell is attracted
to the stable G1 steady state. As it
grows, it eventually passes the
saddle-point bifurcation SN3 where
the G1 steady state disappears.
The cell makes an irreversible
transition into S/G2 until it grows so
large that the S/G2 steady state
disappears, giving way to an infite
period oscillation (SN/IP).
Tyson et al., Curr.Pin.Cell.Biol. 15, 221 (2003)
Cyclin-B-dependent kinase activity soars, driving the cell into mitosis, and then plummets, as cyclin B is degraded by APC–Cdc20. The drop in Cdk1–cyclin B activity is the signal for the cell to divide, causing cell size to be halved from 1.46 to 0.73, and the control system is returned to its starting point, in the domain of attraction of the G1 steady state.