Klipp E., Herwig R., Kowald A., Wierling C., Lehrach H. Systems Biology in Practice: Concepts, Implementation and Application

Подождите немного. Документ загружается.

6.4.3

Negative Feedback

Among the various regulatory features of signaling pathways, negative feedback has

attracted outstanding interest. It also plays an important role in metabolic pathways,

e.g., in amino acid synthesis pathways, where a negative feedback signal from the

amino acid at the end to the precursors at the beginning of the pathway prevents an

overproduction of this amino acid. The implementation of the feedback and the re-

spective dynamic behavior show a wide variation. In Chapter 7 we will show how

feedback can bring about limit cycle–type oscillations in cell cycle models. In signal-

ing pathways, negative feedback may cause an adjustment of the response or

damped oscillations (Section 7.1.1.3).

Example 6-3

Some possible effects of negative feedback are illustrated in Fig. 6.16 for an un-

branched chain of six reactions, as depicted in Eq. (5-190), in which the rate ex-

pressions are v

 S

i1

for i = 1, …, 6. Feedback inhibition of the first reaction

by the j-th metabolite is implemented as v

= S

/(1 + S

In the absence of feedback, the metabolite concentrations reach a steady state

after a short transition period. If the second metabolite inhibits the first reaction,

one may observe an overshooting of the concentration of the first metabolite. In

the case of a long-ranging feedback, oscillations can occur. In the given example,

damped oscillations result from the inhibition of the first reaction by the last me-

tabolite.

6.4.4

Quantitative Measures for Properties of Signaling Pathways

To characterize the dynamic behavior of signaling pathways in a quantitative way,

Heinrich and colleagues (2002) introduced three measures. Let X

(t) be the time-de-

pendent concentration of the kinase i. The signaling time 

describes the average

time to activate the kinase i. The signal duration #

gives the average time during

which the kinase i remains activated. The signal amplitude S

is a measure of the

average concentration of activated kinase i. The following definitions have been in-

troduced. The quantity



tdt (6-27)

is the total amount of active kinase i generated during the signaling period, i.e., the

integrated response of X

(the area covered by a plot X

(t) versus time). Further mea-

sures are

220

6 Signal Transduction



t  X

tdt (6-28)



 X

tdt : (6-29)

The signaling time can now be defined as



 T

; (6-30)

i.e., as the average of time, analogous to mean value of a statistical distribution. Note

that other definitions for characteristic times have been introduced in Chapter 5

(Section 5.2.6). The signal duration







 

(6-31)

221

6.4 Signaling: Dynamic and Regulatory Features

Fig. 6.16 Effect of negative feedback on

the dynamics of a simple unbranched

reaction scheme. Upper panel: no feed-

back. All substrate concentrations reach

asymptotically a high, constant level. Cen-

ter: Feedback from the second metabolite

on first reaction. The substrate concentra-

tions approach a constant but remarkably

lower level. Lower panel: Feedback from

the last metabolite on first reaction. The

substrate concentrations approach a

lower level than without feedback and

show damped oscillations.

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

(compatible to standard deviation). The signal amplitude is defined as

 I



: (6-32)

In a geometric representation, this is the height of a rectangle whose length is 2#

and whose area equals the area under the curve X

(t).

Example 6-4

For the model of the phosphorelay system presented in Section 6.3.3, the charac-

teristic quantities are given in Tab. 6.3. How these measures relate to the actual

time course is shown for Ssk1 in Fig. 6.17.

222

6 Signal Transduction

Fig. 6.17 Representation of characteristic

measures for the output signal of the phos-

phorelay system, Ssk1. The horizontal line

indicates the calculated signal amplitude,

the solid vertical line marks the characteris-

tic time, and the dotted vertical lines cover

the signaling time.

Tab. 6.3 Dynamic characteristics of the phosphorelay system.

Protein Characteristic time Signal duration

Signal amplitude

(s s

–1

)(WW s

–1

)

(A lM

–1

)

Sln1 109.182 57.693 0.151

Ypd1 116.283 51.325 0.170

223

References

Banuett, F. Signalling in the yeasts: an infor-

mational cascade with links to the filamentous

fungi (1998) Microbiol. Mol. Biol. Rev. 62,

249–74

Blumer, K.J. and Thorner, J. Receptor-G pro-

tein signaling in yeast (1991) Annu. Rev. Phy-

siol. 53,37–57

Buck, L.B. The molecular architecture of odor

and pheromone sensing in mammals (2000)

Cell 100, 611–8

Burchett, S.A. Regulators of G protein signal-

ing: a bestiary of modular protein binding do-

mains (2000) J. Neurochem. 75, 1335–51

Dohlman, H.G. G proteins and pheromone sig-

naling (2002) Annu. Rev. Physiol. 64, 129–52

Dohlman, H.G. and Thorner, J. RGS proteins

and signaling by heterotrimeric G proteins

(1997) J. Biol. Chem. 272, 3871–4

Dohlman, H.G. and Thorner, J.W. Regulation

of G protein-initiated signal transduction in

yeast: paradigms and principles (2001) Annu.

Rev. Biochem. 70, 703–54

Dohlman, H.G.,Thorner, J., Caron, M.G.

and Lefkowitz, R.J. Model systems for the

study of seven-transmembrane-segment re-

ceptors (1991) Annu. Rev. Biochem. 60, 653–

Dohlman, H.G., Song, J., Apanovitch, D.M.,

DiBello, P.R. and Gillen, K.M. Regulation of

G protein signalling in yeast (1998) Semin.

Cell. Dev. Biol. 9, 135–41

Force,T., Bonventre, J.V., Heidecker, G.,

Rapp, U., Avruch, J. and Kyriakis, J.M. Enzy-

matic characteristics of the c-Raf-1 protein ki-

nase (1994) Proc. Natl. Acad. Sci. U S A 91,

1270–4

Francke, C., Postma, P.W.,Westerhoff, H.V.,

Blom, J.G. and Peletier, M.A. Why the phos-

photransferase system of Escherichia coli es-

capes diffusion limitation (2003) Biophys. J.

85, 612–22

Ghaemmaghami, S., Huh,W.K., Bower, K.,

Howson, R.W., Belle, A., Dephoure, N.,

O’Shea, E.K. and Weissman, J.S. Global ana-

lysis of protein expression in yeast (2003)

Nature 425, 737–41

Goldbeter, A. and Koshland, D.E., Jr. An am-

plified sensitivity arising from covalent modi-

fication in biological systems (1981) Proc.

Natl. Acad. Sci. USA 78, 6840–4

Goldbeter, A. and Koshland, D.E., Jr. Ultra-

sensitivity in biochemical systems controlled

by covalent modification. Interplay between

zero-order and multistep effects (1984) J. Biol.

Chem. 259, 14441–7

Heinrich, R., Neel, B.G. and Rapoport, T.A.

Mathematical models of protein kinase signal

transduction (2002) Mol. Cell. 9, 957–70

Hohmann, S. Osmotic stress signaling and os-

moadaptation in yeasts (2002) Microbiol. Mol.

Biol. Rev. 66, 300–72

Huang, C.Y. and Ferrell, J.E., Jr. Ultrasensitiv-

ity in the mitogen-activated protein kinase cas-

cade (1996) Proc. Natl. Acad. Sci. USA 93,

10078–83

Kholodenko, B.N. Negative feedback and ul-

trasensitivity can bring about oscillations in

the mitogen-activated protein kinase cascades

(2000) Eur. J. Biochem. 267, 1583–8

Kisseleva,T., Bhattacharya, S., Braunstein,

J. and Schindler, C.W. Signaling through the

JAK/STATpathway, recent advances and

future challenges (2002) Gene 285,1–24

Klipp, E., Nordlander, B., Krueger, R., Gen-

nemark, P. and Hohmann, S. The dynamic

response of yeast cells to osmotic shock – a

systems biology approach (2004) submitted

Meigs,T.E., Fields,T.A., McKee, D.D. and

Casey, P.J. Interaction of Galpha 12 and Gal-

pha 13 with the cytoplasmic domain of cad-

herin provides a mechanism for beta -catenin

release (2001) Proc. Natl. Acad. Sci. USA 98,

519–24

Neer, E.J. Heterotrimeric G proteins: organi-

zers of transmembrane signals (1995) Cell 80,

249–57

Offermanns, S. Mammalian G-protein func-

tion in vivo: new insights through altered

gene expression (2000) Rev. Physiol. Biochem.

Pharmacol. 140, 63–133

Postma, P.W., Broekhuizen, C.P. and Geerse,

R.H. The role of the PEP: carbohydrate phos-

photransferase system in the regulation of

bacterial metabolism (1989) FEMS Microbiol.

Rev. 5,69–80

Postma, P.W., Lengeler, J.W. and Jacobson,

G.R. Phosphoenolpyruvate:carbohydrate phos-

photransferase systems of bacteria (1993) Mi-

crobiol. Rev. 57, 543–94

Rohwer, J.M., Meadow, N.D., Roseman, S.,

Westerhoff, H.V. and Postma, P.W. Under-

standing glucose transport by the bacterial

phosphoenolpyruvate:glycose phosphotrans-

ferase system on the basis of kinetic measure-

224

6 Signal Transduction

ments in vitro (2000) J. Biol. Chem. 275,

34909–21

Ross, E.M. and Wilkie, T.M. GTPase-activating

proteins for heterotrimeric G proteins: regula-

tors of G protein signaling (RGS) and RGS-

like proteins (2000) Annu. Rev. Biochem. 69,

795–827

Schindler, C.W. Series introduction. JAK-

STATsignaling in human disease (2002) J.

Clin. Invest. 109, 1133–7

Schmidt, A. and Hall, A. Guanine nucleotide

exchange factors for Rho GTPases: turning on

the switch (2002) Genes Dev. 16, 1587–609

Siderovski, D.P., Strockbine, B. and Behe,

C.I. Whither goest the RGS proteins? (1999)

Crit. Rev. Biochem. Mol. Biol. 34, 215–51

Swameye, I., Muller,T.G., Timmer, J.,

Sandra, O. and Klingmuller, U. Identifica-

tion of nucleocytoplasmic cycling as a remote

sensor in cellular signaling by databased mod-

eling (2003) Proc. Natl. Acad. Sci. USA 100,

1028–33

Takai,Y., Sasaki,T. and Matozaki, T. Small

GTP-binding proteins (2001) Physiol. Rev. 81,

153–208

Tyson, J.J., Chen, K.C. and Novak, B. Sniffers,

buzzers, toggles and blinkers: dynamics of

regulatory and signaling pathways in the cell

(2003) Curr. Opin. Cell. Biol. 15, 221–31

Wang, P. and Heitman, J. Signal transduction

cascades regulating mating, filamentation,

and virulence in Cryptococcus neoformans

(1999) Curr. Opin. Microbiol. 2, 358–62

Wilkinson, M.G. and Millar, J.B. Control of

the eukaryotic cell cycle by MAP kinase signal-

ing pathways (2000) Faseb J 14, 2147–57

Yi,T.M., Kitano, H. and Simon, M.I. A quanti-

tative characterization of the yeast heterotri-

meric G protein cycle (2003) Proc. Natl. Acad.

Sci. USA 100, 10764–9

Selected Biological Processes

Introduction

Biological processes that are of special interest for systems biology are those that

contain many reactions forming a complex system or that have the potential to inter-

act and influence many other processes. In this chapter we will take a closer look at

three selected processes that fulfill these requirements, namely, biological oscilla-

tions, the cell cycle, and the aging process. Oscillations represent a ubiquitous phe-

nomenon in biology that often drives other complex and important functions of the

organism. An example of such a central periodic function is the cell cycle, which

dominates many aspects of cellular biochemistry. Furthermore, modifications to cri-

tical components of the cell cycle can lead to diseases such as cancer or Alzheimer’s

disease. The final example, aging, is a complex phenomenon that affects practically

all components of an organism and becomes ever more important as the average life

span of the world population increases. Because of space limitations we restricted

ourselves to the mentioned examples, but of course there are other candidates (for

instance, development or the immune system) that in principle would fit into this

section.

7.1

Biological Oscillations

Periodic changes of biochemical and biophysical quantities are a universal phenom-

enon in living systems. Examples from everyday experience are the pulse of the

heart, spontaneous respiration, the circadian rhythm, cycles of ovulation in mam-

mals, or the annual flowering of trees. Well studied are calcium waves (Goldbeter

et al. 1990; Bootman et al. 2001a, 2001 b), oscillations in neuronal signals (Rabino-

vich and Abarbanel 1998), oscillations in cyclic AMP in the slime mold Dictyostelium

discoideum (Roos et al. 1977; Halloy et al. 1998; Nanjundiah 1998), the periodic con-

version of sugar to alcohol (glycolysis) in anaerobic yeast cultures (Chance et al.

1964; Ghosh and Chance 1964; Sel’kov 1968), the circadian rhythm (Smaaland 1996;

Turek 1998), and the cell cycle (Tyson 1991; Tyson et al. 1995; Novak et al. 1999;

Mori and Johnson 2000; Tyson and Novak 2001). Periodic patterns can be a function

225

of time (glycolytic oscillations), space (striping in Drosophila melanogaster embryos),

or both (D. discoideum, calcium waves, neuronal oscillations), depending on the me-

chanism of the oscillator. The oscillation periods may cover ranges from millise-

conds to years. Some oscillations are initiated externally, while others have intrinsic

causes. Many cellular oscillations are associated with the regulation of enzyme activ-

ity, receptor function, transport processes, or gene expression in an autocatalytic

manner or by positive or negative feedback and feed-forward loops. Other cases of

oscillations arise from the regulation of ionic conductances in electrically excitable

cells.

Temporarily changing patterns are observed in different complexity. Types of beha-

vior include simple periodic oscillations, complex periodicity with several maxima

per period, and even irregular and aperiodic behavior, owing to the appearance of

chaos. In the following sections we will introduce the Higgins-Sel’kov oscillator as a

classical example of oscillations caused by positive feedback and a model of a multi-

ply regulated biochemical system as an example of more complex oscillatory pat-

terns. Coupled oscillators are presented to illustrate that oscillations in individual

cells are sometimes hidden on the population level and therefore are hard to mea-

sure experimentally.

7.1.1

Glycolytic Oscillations: The Higgins-Sel’kov Oscillator

The product-activated enzyme reaction is a simple model with two variables for peri-

odic oscillations of the limit-cycle type. The most intensively studied example is the

positive feedback exerted by ADP on the enzyme phosphofructokinase I (PFK I). It is

believed to cause the oscillations observed in glycolysis in yeast and muscles. The dy-

namic behavior of this model system was first studied by Higgins (1964) and Sel’kov

(1968) and later by many others (e. g., Goldbeter and Lefever 1972; Sel’kov 1975).

The two-variable model takes into account the allosteric regulation of the enzyme

PFK I and the autocatalytic effect exerted by the product. For a large range of para-

meter values, it exhibits a stable steady state, but beyond a critical parameter value,

the system becomes instable and evolves towards a stable limit cycle. Then, it shows

sustained oscillations.

(7-1)

The temporal behavior of the concentrations of substrate S and product P can be

described by the following ODEs:

 v

 Sk

 rP

 Sk

 rPPk

: (7-2)

226

7 Selected Biological Processes

The supply rate of S, v

, is positive. The parameters k

, k

are mass-action rate con-

stants. The function r(P) represents the autocatalytic effect of the product P on its

own production. The simplest expression for this function is

rPP

; (7-3)

yielding

 v

 SP

 fS; P

 SP

 Pk

 gS; P: (7-4)

The dynamic behavior of this system is represented in Fig. 7.1 for a set of para-

meters that gives rise to oscillations. Figure 7.1a shows the values of the variables as

function of time. Further information about the dynamics of the system can be in-

227

7.1 Biological Oscillations

Fig. 7.1 Higgins-Sel’kov oscillator. The dy-

namic behavior of the ODE system in Eq. (7-4)

is presented for the following choice of para-

meters: v

=1,k

= 1.00001, S(0) = 2,

and P(0) = 1 in arbitrary units. (a) Time course

of the solution. (b) Stability regions for fixed

values of k

= 1 and varying values of v

and k

of the solution. (d) Representation of nullclines

dS/dt = 0 (black line) and dP/dt = 0 (gray line) in

the phase plane. Arrows indicate the direction of

a trajectory that crosses the nullcline at the re-

spective point.

ferred by inspection of the phase plane. Figure 7.1 c shows the trajectory for a given

set of parameters and initial conditions in a plot of S versus P, using the time as

parameter. The nullclines, i.e., the lines for dS/dt = f =0ordP/dt = g = 0, respectively,

are shown in Fig. 7.1d. These lines must always be crossed by the trajectories in a

horizontal (for f = 0) or vertical (for g = 0) manner, respectively, as indicated by the

little arrows. The sign of g determines the direction of the arrow for the nullcline

f = 0 at a certain point, and vice versa.

The steady state of the equation system in Eq. (7-4) is unique and is determined by



S 

;



P 

: (7-5)

The stability of the steady state can be analyzed by inspection of the Jacobian ma-

trix (Chapter 3.2),

J 





2



 k



v

2 k

: (7-6)

The character of the steady state is given by the determinant and the trace of the

Jacobian matrix. The determinant reads

DetJ  v

: (7-7)

Since v

> 0, the determinant is always positive. However, the trace

TraceJ v

 k

(7-8)

changes its sign at



: (7-9)

This surface separates stable from unstable steady states in the parameter space.

Further critical values can be found by the condition (TraceJ)

=4DetJ (Section 3.2.3)

separating nodes from foci. This condition is fulfilled at

 6 v

 k

 0 : (7-10)

At the transition from the region of stable focus ((TraceJ)

<4Det, TraceJ < 0) to

the region of instable focus ((TraceJ)

<4Det, TraceJ > 0), limit cycles arise. The

228

7 Selected Biological Processes

change in the stability and character of the steady state with changing parameters is

called a bifurcation. This special type of bifurcation, where the eigenvalues of the

Jacobian matrix are purely imaginary, is called a Hopf bifurcation (Hopf 1942). The

stability regions are represented in Fig. 7.1b.

It should be noted that the different types of dynamics occur upon changes in the

system parameters. It is convenient to keep some of the parameters constant and to

follow the change of one parameter, in order to observe the regions of various beha-

viors. If, for example, k

and v

are kept constant, then the value of the kinetic con-

stant of product degradation, k

, is an indicator for the dynamic behavior and can

mark the point of the Hopf bifurcation.

7.1.2

Other Modes of Behavior

In biological systems, dynamics have been observed that are more complex than just

simple oscillation with fixed amplitudes and frequencies. An example of this type of

dynamic behavior is a system of two product-activated enzyme reactions that are

coupled in series.

(7-11)

The system involves three variables: the substrate S and the products P

and P

S is produced, the two enzymes E

and E

transform S into P

and P

into P

, respec-

tively, and P

is degraded. Their respective products activate both allosteric enzymes.

Thus we now have two positive feedback loops. Each of the reactions may show in-

stability as discussed above.

Decroly and Goldbeter (1982) investigated intensively the possible modes of beha-

vior of this system, assuming that the kinetics of both enzymes obeys Monod ki-

netics (Chapter 5, Eqs. (5-60) and (5-61)). The respective equation system reads



 s



S 1 S1 P



 1 S

1  P



 q

 s



S 1 S1 P



 1 S

1  P



 s



1 d  P

1  P



 1 d  P



1  P



 q

 s



1 d  P

1  P



 1 d  P



1 P



 k

; (7-12)

where K

is the Michealis constant for the substrate, s

is the maximal activity, L

the allosteric constant of enzyme i, and q

and d are ratios of Michaelis constants. In

the investigation of the dynamic behavior, all parameters are kept fixed except for the

rate constant k

for the degradation of P

. It could be shown that for increasing

values of k

the system may assume (1) a stable steady state; (2) a limit cycle with

229

7.1 Biological Oscillations