v14 dynamic cellular processes

14. Lecture WS 2007/08

Bioinformatics III 1

V14 Dynamic Cellular Processes

John Tyson Bela Novak



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



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 S.

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).

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.



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




cell-cycle machineryThe signalling dynamics can become multi-stable when two or more bistable cycles form a

cascade, such as a MAPK cascade. The biological outcome of multistability is the ability to

control multiple irreversible transitions, for instance, sequential transitions in the cell cycle.

Central components of the cell-cycle machinery are cyclin-dependent kinases (such as CDK1/ CDC2), the sequential activation and inactivation of which govern cell-cycle transitions. The activity of CDK1/CDC2 is low (off) in the G1 phase and has to be high (on) for entry into mitosis (M phase). Tyson et al., Curr.Opin.Cell.Biol. 15, 221 (2003)





signal: synthesis of Cdk1:CycB

response: Cdk1/CycB

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 infinite

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.



cell-cycle machineryHysteresis and bistability were recently shown to occur in the activation/ inactivation of

CDK1/CDC2, an observation that confirmed a theoretical prediction by Novak and Tyson 10

years ago.

Bistability in the CDK1/CDC2 cycle arises from positive and double-negative feedback loops in

the reactions. CDK1/CDC2 activates its activator (the phosphatase CDC25) and inactivates its

inhibitors (the kinases Wee1 and Myt1).

Negative feedback from the anaphase-promoting complex (APC) turns the CDK1/CDC2

bistable switch into a relaxation oscillator that drives the cell cycle.

Intriguingly, CDC25 and Wee1 can be phosphorylated on multiple sites and can therefore

potentially exhibit bistability, which implies that the entire CDK/cyclin system can display

multiple steady states — this prediction is awaiting experimental verification.

Sequential bifurcations of multiple steady states provide more flexibility in the control of the cell

fate and allow for several checkpoints in the cell cycle.



V17 Modelling signalling cascades

Cells respond to external stimuli using a limited number of signalling pathways that

are activated by plasmamembrane receptors, such as G protein-coupled receptors

(GPCRs) and receptor tyrosine kinases (RTKs).

These pathways do not simply transmit, but they also process, encode and integrate

internal and external signals.

Distinct spatio-temporal activation profiles of the same repertoire of signalling

proteins may´result in different gene-expression patterns and diverse physiological

responses pivotal cellular decisions, such as cytoskeletal reorganization, cell-

cycle checkpoints and cell death (apoptosis), depend on the precise temporal

control and relative spatial distribution of activated signal transducers.

B.N. Kholodenko, Nature Rev. Mol. Cell. Biol. 7, 165 (2006)



receptor tyrosine kinasesRTK-mediated signalling pathways have a central role in the regulation of embryogenesis,

cell survival, motility, proliferation, differentiation, glucose metabolism and apoptosis.

Malfunction of RTK signalling is a leading cause of important human diseases involving e.g.

developmental defects, cancer, chronic inflammatory syndromes and diabetes.

Upon stimulation, RTKs undergo dimerization (e.g. the epidermal growth factor receptor.

EGFR) or allosteric transitions (insulin receptor) that result in the activation of the intrinsic

tyrosine-kinase activity.

Subsequent phosphorylation of multiple tyrosine residues on the receptor transmits a

biochemical signal to numerous cytoplasmic proteins, thereby triggering their mobilization to

the cell surface.

The resulting cellular responses occur through complex biochemical circuits of protein–

protein interactions and covalent-modification cascades.



receptor tyrosine kinases

Earlier concept: discrete linear pathways that relate extracellular signals to the

expression of specific concepts.

New concept: interconnected signalling networks. Several triggers use the same

pathway in different fashions to achieve different output signals. How?

E.g. PC12 cell-line stimulation with epidermal growth factor (EGF) and nerve

growth factor (NGF). Both stimulate the MAPK cascade.

EGF induces transient MAPK activation which results in cell proliferation.

NGF creates a sustained MAPK activation that changes the cell fate and induces

cell differentiation.



Cycle and cascade motifs

A universal motif that is

found in cellular networks is

the cycle that is formed by

two or more interconvertible

forms of a signalling protein.

This protein is modified by

two opposing enzymes.

These are a kinase and a

phosphatase for phospho

proteins,

and a guanine nucleotide

exchange factor (GEF) and a

GTPase-activating protein

(GAP) for small G proteins.



Feedback loops induce complex dynamicsFeedback is one of the most fundamental concepts in biological control. An increase in the number of interconnecting cycles in a cascade or positive feedback further increases the sensitivity of the target to the input signal.

Positive feedback amplifies the signal, whereas negative feedback attenuates it. However, feedback loops not only change steady-state responses, but also favour the occurrence of instabilities. When a steady state becomes unstable, a system can jump to another stable state, start to oscillate or exhibit chaotic behaviour.

Positive feedback can cause bistability. Furthermore, positive feedback, alone or in combination with negative feedback, can trigger oscillations; for example, the Ca2+ oscillations that arise from Ca2+-induced Ca2+ release and the cell-cycle oscillations.Such positive-feedback oscillations generally donot have sinusoidal shapes and are referred to as relaxation oscillations, operating in a pulsatory manner: a part of a dynamic system is bistable, and there is a slow process that periodically forces the system to jump between ‘off ’ and ‘on’ states, generating oscillations.



complex dynamics

Complex dynamic properties have traditionally been associated with cascades

of cycles. Yet, even single cycles can exhibit complex dynamics, such as

bistability and relaxation oscillations (see previous slide).

A simple one-site modification cycle can turn into a bistable switch by four

different regulatory mechanisms, in which one of the protein forms stimulates its

own production or inhibits its consumption, thereby creating a destabilizing control

loop.

An extra (stabilizing) feedback loop that affects the rate of synthesis or

degradation of a converting enzyme can render this bistable switch into a

relaxation oscillator (32 distinct feedback designs result that can give rise to

oscillations).



receptor tyrosine kinasesFeedback designs that can turn a universal signalling cycle into a bistable switch and relaxation oscillator. A simple cycle can turn bistable in 4 distinct ways: either a protein M or its phosphorylated form Mp stimulates its own production (positive feedback) by product activation or substrate inhibition of the kinase (Kin) or phosphatase (Phos) reactions.

Each of the 4 rows of feedback designscorresponds to a different bistable switch, provided that the kinase and the phosphatase abundances are assumed constant and only a single feedback (within the M cycle) is present.

Sixteen relaxation-oscillation designs are generated by extra negative feedback brought about by negative or positive regulation of the synthesis or degradation rates of the kinase protein or phosphatase protein by M or Mp. Designs a*–h* are mirror images of designs a–h.

Although synthesis and degradation reactions are shown for both the kinase and the phosphatase proteins, the protein concentration that is not controlled by feedback from the M cycle is considered constant, therefore it results in only two differential equations for each diagram. All the feedback regulations are described by simple Michaelis–Menten-type equations.



receptor tyrosine kinases



Signal Transduction (II)

Study receptor-stimulated kinase/phosphotase signaling cascades as a model

case of signal transduction networks.

- amplitude of the signal output

- rate and duration of signaling



Signal Transduction (II)To simplify the analysis, we first consider a simple linear

signaling cascade in which stimulation of a receptor leads

to the consecutive activation of several downstream

protein kinases (Figure 1).

The signal output of this pathway is the phosphorylation of

the last kinase which, in turn, can elicit a cellular response

(e.g., activation of a transcription factor).

Signaling is terminated by phosphatases, which

dephosphorylate the kinases, by inactivation of the

receptor, which can involve receptor dephosphorylation,

internalization of the receptor ligand complex, and/or

degradation of the receptor or ligand.

This general scheme is representative of many signaling

pathways. E.g., growth factors such as EGF, PDGF, or

NGF stimulate a receptor tyrosine kinase (RTK), which

leads to the activation of three or four consecutive downstream kinases (e.g., Raf, MEK, ERK, and RSK).

Growth factor signals are terminated by protein-tyrosine

phosphatases, RTK endocytosis and degradation, protein

serine-threonine phosphatases, and dual-specificity and

tyrosine-specific MAP kinase phosphatases. The same

type scheme can model pathways that include lipid

kinases, such as PI3K, whose reaction products help to

activate downstream kinases such as PDK1 and Akt.



Linear Signalling Cascades

Describe each phosphorylation step as a reaction between the

phosphorylated form Xi-1 of kinase i – 1 and

the non-phosphorylated form Xi of a downstream kinase i.

Phosphorylation rate: second-order rate constant (2 proteins involved):

~

iiiipXX~~

1,

It is assumed that the concentration of each kinase-substrate complex is small

compared with the total concentration of the reaction partners.

Assuming that the concentration of active phosphatase is constant,

dephosphorylation can be modeled as a first order reaction:

iiidX

,




For all but the first activated kinase in the pathway, the concentration of each

activated kinase i as a function of time, Xi(t) is:

1 ,~~

1,,

iXXX

dt

dXiiiiiidip

i

Define as the total concentration of kinase i and

as a pseudo-first order tate constant. The above equation becomes

iiiXXC ~

iiiC ~

ii

i

i

ii

i XC

XX

dt

dX

1

1

For the first kinase (X1), activation occurs via the stimulated receptor, and

inactivation is mediated by phosphatase 1. Therefore, we have instead

11

1

1

1

1 1 XC

XtR

dt

dX




Assume that all pathway components are inactive at the start and they undergo

rapid stimulation by setting the concentration of active receptor at t = 0 to R.

Model receptor inactivation by

3 key questions are:

(1) How fast does the signal arrive at its destination?

(2) How long does the signal last?

(3) How strong is the signal?

ttR exp



Signalling time

The signalling time i is the average time to activate kinase i.

0

0

dtttXT

dttXI

I

T

ii

ii

i

i

i

Ii, the integrated response of Xi is

the total amount of active kinase i

generated during the signaling

period.

Ii corresponds to the area under the

curve of Xi versus time.

Ti / Ii is an average, analogous to

the mean value of a statistical

distribution.



Signal duration

The signal duration is given by

0

22 where, dttXtQI

Qiii

i

i

i

i gives a measure of how extended the signaling response is around the mean

time.

The signaling amplitude Si is given by

In a geometric representation, Si is the height of a rectangle whose length is 2i

and whose area equals the area under the curve Xi(t).

i

i

i

IS

2



Weakly activated pathways

A pathway is termed „weakly activated“ if all its component kinases are

phosphorylated to a low degree (Xi << Ci).

This may occur when the concentration of activated receptor is low, when the

receptor is rapidly inactivated, and/or when the kinases are present at high

concentrations.

and the key parameters can be calculated explicitly. E.g.

ii

i

i

ii

i XC

XX

dt

dX

1

1 iiii

i XXdt

dX 1

2 ,

11

0

12

2

10 R

SS

Sn

jj

n

kk

k

S depends on the kinetic properties of all pathway components.

High signal amplitudes are obtained with fast kinases and slow phosphatases.



Weakly activated pathways

n

jj

n

jj

n

jj

n

kk

kSS

122

1

12

2

10 11

,11

,1

1

The signaling time and duration do not depend on the kinase rate constants.

Thus, in a weakly activated pathway, kinases regulate only signal amplitude, not

signaling time or duration. In contrast, phosphatases affect all of these

parameters, and in the same direction.



Amplification and dampening

n

jj

n

kk

kSS

12

2

10

11

At any step in a signaling cascade, a signal can be amplified (Si > Si-1),

dampened (Si < Si-1), or remain constant. Amplification at step i will occur if

2

1

2

11

ii

ii

Amplification at step i requires that the phosphate rate constant i for a given

reaction is small compared to the kinase rate constant i.

Amplification at step i also depends on the signal duration at the preceding step

(i-1). When the signal duration was long, amplification can be achieved even

when the phosphatase rate constant at step i is high (but still smaller than i).



Amplification and dampening



Summary

Kinetic modelling of signal transduction networks by systems of coupled

ordinary differential equations:- challenge consists of deriving the appropriate sets of equations- solving their time-dependent behavior is routine when done numerically,

may be very challenging (or impossible) if done analytically.

Other approaches for „data-driven“ modelling of signal-transduction networks:- clustering- principal component analysis- partial least squares regression

See e.g. Janes & Yaffe, Nat. Rev. Mol. Cell. Biol. 7, 820 (2006)

v14 dynamic cellular processes

Documents

negative feedback loop

negativefeedback control

synthesis of r

response element r

steadystate response

component system x r

r removal

s decreases