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)



The Cyclin – E2F cell cycle control system

as

Kohn, Molec. Biol. Cell 1999 10:2703-34



Biocarta pathways

http://cgap.nci.nih.gov/Pathways/BioCarta/h_mapkPathway



Ras signaling pathway

http://cgap.nci.nih.gov/Pathways/BioCarta/h_mapkPathway



Biocarta pathwaysIncredible amount

of information about

signalling pathways!

Constantly updated.

How can we digest this?

Need computational

models!



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



Protein synthesis and degradation

RkSkkdt

dR210


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



phosphorylation/dephosphorylation: hyperbolic response


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



phosphorylation/dephosphorylation: sigmoidal response


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



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



perfect adaptation: sniffer


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“.



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



Positive feedback: Mutual activation


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



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)



mutual inhibition


Here, R inhibits E, and E promotes the degradation of R.

4343

'

2210

,,, JJRkkGRE

RREkRkSkkdt

dR



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)



Negative feedback: homeostasis


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.



Negative feedback: oscillatory response


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



Negative feedback: oscillatory response

Feedback loop leads to

oscillations of X (black),

YP (red), and RP (blue).


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.



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.


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.



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.




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.




Cell cycle control system






The G1/S module is a toggle switch, based on mutual inhibition between

Cdk1-cyclin B and CKI, a stoichiometric cyclin-dependent kinase inhibitor.





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).





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.




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).


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.

v22 modelling dynamic cellular processes

Documents

response component r

signal duration

response elementrp

response magnitude

transient response

high signal amplification

function of signal strength

magnitudes of signal