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trate this, consider Eq. (7-31), which describes the time course of intact mitochondria

with little membrane damage (M

). The first term defines the increase of M

caused by synthesis of new mitochondria. The next term describes the fraction of mi-

tochondria belonging to M

M2,

which are converted into M

because of the membrane

damage dilution effect (see previous section). The last term summarizes all processes

leading to a loss of M

, i.e., decay (with rate constant a) and acquisition of further

membrane (k

) or DNA damage (k

). Similar terms exist in the other equations such

as in Eq. (7-35) for M

DM2

. The individual expressions describe loss and gain caused by

the damage dilution process, increase through M

that suffered DNA damage and

DM1

that acquired further membrane damage, and finally the loss caused by degra-

dation and membrane damage. In some of these terms, the expression RDF 7Rad

appears. This reflects the fact that the amount of radicals in damaged mitochondria

can be calculated from Rad

by multiplication with the factor RDF.

Equation (7-37) for the antioxidants consists of just two expressions describing

synthesis and degradation. Synthesis is controlled by the ATP level and the antioxi-

dant promoter. As for mitochondria, it is assumed that a low level of ATP diminishes

the protein synthesis rate.

In the model it is assumed that the removal of radicals, Eq. (7-38), depends on the

amount of antioxidants per mitochondrion. Although this is a reasonable assump-

tion, other possibilities, such as elimination depending on the total amount of anti-

oxidants, exist. In a qualitative model like this, the exact molecular details of the

modeled reactions are often not known, and thus neither way of modeling would be

wrong. If in doubt, both versions should be implemented and the simulation results

compared.

Regarding the generation of energy, Eq. (7-39), it is assumed that ATP production

is directly proportional to the proposed growth disadvantage. This means that M

generates half of the energy of M

and that all other types of mitochondria do not

contribute to the cellular energy production. The model incorporates three energy-

consuming processes: the synthesis of mitochondria, the synthesis of antioxidants,

and a fixed amount k

that represents all other energy-consuming processes of the

cell that are not explicitly included in the model.

 S  M



GDF  1

 M

 a k

 k

Rad

M

(7-31)



GDF  1

 M



GDF

 M

 k

Rad

 M

 b k

 k

Rad

M

(7-32)



GDF

 M

 k

Rad

 M

 g  k

Rad

M

(7-33)

DM1



GDF

 M

DM1

 M

DM2

k

Rad

 M

 ak

 RDF  Rad

M

DM1

(7-34)

250

7 Selected Biological Processes

DM2



GDF

 M

DM2



GDF

 M

DM3

k

Rad

M

k

RDF Rad

 M

DM1

   k

 RDF  Rad

M

DM2

(7-35)

DM3



GDF

 M

DM3

 k

Rad

 M

 k

 RDF  Rad

 M

DM2

 g  M

DM3

(7-36)

dAOx



ATP

ATP ATP



1 

PAOx

Rad

M

RDF  Rad

DM1

 M

DM2

M

DM3





d AOx

(7-37)

dRad

 k

 k

AOx  Rad

 M

DM1

 M

DM2

 M

DM3

(7-38)

dATP

 k

ATP

 M



ATP

 M



ATP

ATP  ATP



1 

ATP



 k



1 

PAOx

Rad

M

RDF  Rad

DM1

M

DM2

M

DM3





(7-39)

S 

ATP

ATP ATP



1 

ATP







GDF  1

 M



GDF

 M

 M

DM1

 M

DM2

 M

DM3



: (7-40)

7.3.2.5 Choice of Parameters and Simulation Results

The numerical values of the model parameters can have a large influence on the dy-

namic behavior of the model. Ideally, all parameters used by the model would have a

clear biochemical representation and all parameters would be measured experimen-

tally, so that these values can be used directly for the numerical simulation. Unfortu-

nately, this situation only rarely applies. Often a biological process that in reality con-

sists of several reactions is collapsed into a single reaction because the details are

either unknown or of no importance for the current model. An example is the gen-

eric antioxidant (AOx) that is used in this model to represent all cellular antioxidant

reactions. Clearly, the parameters controlling the synthesis, degradation, or activity

of AOx have no direct presentation in reality. In this case an educated guess has to

251

7.3 Aging

be made to choose initial parameter settings. Additionally, a parameter scan should

be performed to see whether the model behavior depends critically on the chosen

numerical values. The other reason that exact numerical values cannot be used is of

course that they have not yet been measured (or cannot be measured for technical

reasons). GDF, RDF, k

, k

, and k

are examples of such parameters. A qualita-

tive model can cope with uncertain parameters by varying those parameters over a

possible range of values to see if this leads to a qualitative change in the model pre-

dictions (e.g., loss of stable steady state, appearance of oscillations, drastic changes

in timescale). Quantitative models that seek to predict or reproduce experimental

data as closely as possible can try to estimate the unknown parameters by fitting the

model to experimental measurements. This, however, depends on the availability of

such data.

The following table lists the parameters of the SOS model together with the stan-

dard values used for the numerical simulations and a short description.

Name Value Description

a 0.01 d

–1

Decay rate for mitochondria with little membrane damage (M

, M

DM1

The resulting half-life is 70 days. The values of a, b, and g are based on the

data of Huemer et al. (1971) and Menzies and Gold (1971).

b 0.05 d

–1

Decay rate for mitochondria with medium membrane damage (M

, M

DM2

The resulting half-life is 14 days.

g 0.1 d

–1

Decay rate for mitochondria with much membrane damage (M

, M

DM3

The resulting half-life is 7 days.

d 0.693 d

–1

Decay rate for antioxidants. The resulting half-life is 1 day.

RDF 0.2 Factor specifying how much fewer radicals are generated by mutant mito-

chondria than by intact ones.

GDF 5 Factor specifying the growth rate difference between intact and mutant mito-

chondria.

0.003 d

–1

Constant controlling the rate of membrane damage accumulation.

0.003 d

–1

Constant controlling the rate of mtDNA damage accumulation.

100 d

–1

Maximum number of mitochondria synthesized per day.

100 d

–1

Maximum number of antioxidants synthesized per day.

7000 d

–1

Rate of radical removal by antioxidants in units of 10

. This value is based on

work of Rotilio et al. (1972) concerning superoxide dismutase.

400 Molecules of ATP needed for the synthesis of a mitochondrion in units of

. It is assumed that the main costs are synthesizing lipids and proteins in

the membrane and the proteins of the matrix. To calculate these costs, the

following assumptions have been made. A mitochondrion is a rod-like struc-

ture 0.5 mm in diameter and 1 mm long. The matrix consists of 18% proteins

with an average weight of 40,000 Da. The membrane consists of 75% pro-

tein, which is equivalent to 25 lipid molecules per protein. The diameter of

a lipid is approximately 1.5 nm and 5.5 nm for a protein. Finally, 800 mole-

cules of ATP are required to synthesize one protein and 15 molecules to

synthesize one lipid. These assumptions are based on the data of Alberts

et al. (1994) and Stryer (1988).

0.0008 Molecules of ATP needed for the synthesis of one antioxidant protein in

units of 10

. A size of 400 amino acids is assumed.

252

7 Selected Biological Processes

Name Value Description

–1

Amount of ATP consumed per day per cell in units of 10

. This value is cal-

culated under the assumptions that the specific metabolic rate for humans is

0.25 mL of O

per g/h (Adelman et al. 1988), that six molecules of ATP are

generated from each molecule of oxygen, and that the cell volume is

1000 mm

900 d

–1

Radicals produced by genetically intact mitochondria per day in units of 10

It is assumed that 10

radicals are generated per second per cell (Joenje et al.

1985) and that the cell has a mitochondrial population of approximately 100.

ATP

1200 d

–1

ATP produced per day by a functional mitochondrion (M

) in units of 10

ATP

100 Promoter control parameter for the synthesis of mitochondria. If the ATP le-

vel is equal to ATP

, the promoter is 50% repressed.

PAO

1 Promoter control parameter for the synthesis of antioxidants. If the radical

level is equal to PAO

, the promoter is 50% activated.

After all of the efforts to develop the equations for the model of the SOS hypoth-

esis Fig. 7.12 finally shows some numerical results that have been obtained by sol-

ving the set of coupled ODEs with Mathematica (see Section 14.1.1). For the simula-

tion shown on the left side, the standard values of the decay rates a, b, and g were

used and the system quickly reached a stable steady state. The simulation on the

right side investigates how the resulting steady-state values are influenced by

changes in the turnover rate. For convenience, the turnover rate is shown here in

multiples of the standard turnover rate. This means that the time course shown in

Fig. 7.12 a approaches the steady-state values corresponding to a turnover rate of one

253

7.3 Aging

intact mito.

defect mito.

radicals

ATP

Time

0 100 200 300 400 500

100

200

300

400

500

600

700

800

200

400

600

800

1000

024681012

Turnover rate (arbitrary units)

Steady state concentrations

radicals

intact mito.

defect. mito.

ATP

ATP consumption

Fig. 7.12 Numerical solution of the model

equations. To display all variables in one dia-

gram, appropriate scaling factors have been ap-

plied to the different variables. (a) For this calcu-

lation, the standard values for the mitochondrial

turnover rates a, b, and g were used. In this

case, the population quickly reaches a stable

steady state. (b) Parameter scan showing how a

variation of the turnover rate (in multiples of the

standard values) affects the steady-state levels.

The important conclusion is that a turnover rate

that is either too low (<0.6) or too high (>11)

leads to an unstable system.

in Fig. 7.12b. If the turnover rate is too low (< 0.6) or too high (>11), the system has

no stable steady state, which means that the mitochondrial population will finally

break down. If the turnover rate is too high, mitochondria are degraded faster than

they can be synthesized by the cellular machinery. But, if the turnover rate is too low,

mitochondria have a much longer time to accumulate damage, because now the

half-life of the organelles is substantially increased. This additional burden also leads

to a collapse of the system.

With such a complex model as we have developed here, there are of course many

more simulations possible. It is especially possible to investigate the consequences

of cell division on the stability of the mitochondrial population. However, for further

details, the reader is referred to Kowald and Kirkwood (1999, 2000).

254

7 Selected Biological Processes

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Modeling of Gene Expression

Introduction

The expression of genes, which is a highly regulated process in eukaryotic as well as

in prokaryotic cells, has a profound impact on the ability of the cells to maintain

vitality, perform cell division, and respond to environmental changes or stimuli. In

theoretical modeling of gene expression, two diverse approaches have been devel-

oped. On the one hand, the expression of one or a few genes has been described on

the level of transcription or translation by detailed mathematical models that include

the binding of transcription factors and RNA polymerases to DNA, the effect of spe-

cific inhibitors or activators, the formation of various stages of maturation of mRNA

or proteins, and the regulation by internal feedback loops or external regulators. The

basis of this type of modeling is knowledge or hypotheses about the processes and

interactions taking place during gene expression. Like most types of kinetic model-

ing, it often lacks specific kinetic parameters for the individual processes under con-

sideration. On the other hand, the expression changes of thousands of genes are ana-

lyzed in parallel over time with DNA arrays. Gene expression profiles (expression

levels at different time points) and gene expression patterns (comparison of expres-

sion values of different genes under different experimental conditions) are used to

search for clusters and motifs and, eventually, to deduce functional correlations.

Based on this information, reverse engineering methods seek to reconstruct the

underlying regulatory networks (Section 9.6). While these approaches largely neglect

the highly complex regulatory machinery behind the emergence of detectable

mRNA involving the action of proteins and other regulatory molecules, they cover a

large fraction or almost all genes of a cell – compared to the first approach, which

can deal only with a few genes.

In this chapter we describe the basic steps of gene expression (Section 8.1). A fun-

damental step is the regulation of transcription through transcription factors that

bind to DNA motifs upstream of the transcription start point. Section 8.2 introduces

computational methods for the detection of binding sites that determine such tran-

scription factor binding to the DNA. Section 8.3 covers the detailed modeling of

basic processes of eukaryotic gene expression, and Section 8.4 discusses the regula-

tion of gene expression in prokaryotes in the context of the operon concept.

257

8.1

Modules of Gene Expression

In the following section, we will outline a general view of the processes representing

gene expression, from the activation of transcriptional regulators to the synthesis of

a functional protein (Orphanides and Reinberg 2002; Proudfoot et al. 2002; Reed

and Hurt 2002).

Hundreds of different cell types exist and fulfill specific roles in the organism.

Each cell type theoretically contains information on the same set of genes; however,

only a proportion of these genes is expressed, determining the specific role of cells

of this type. Gene expression in eukaryotes is controlled at six different steps, which

determine the diversity and specification of the organism (Alberts et al. 2002):

1. Transcriptional control: when and how often a gene is transcribed.

2. RNA processing control: how the RNA transcript is spliced.

3. RNA transport and localization control: which mRNAs in the nucleus are ex-

ported to cytosol and where in the cytosol they are localized.

4. Translation control: which mRNAs in the cytosol are translated by ribosomes.

5. mRNA degradation control: which mRNAs in the cytosol are destabilized.

6. Protein activity control: determines activation, inhibition, compartmentalization,

and degrading of the translated protein.

Each step is complex and has been studied extensively in isolation. The process is

typically modeled with a linear structure of more or less independent modules where

the output of the previous module is the input for the current module.

The expression level of the majority of genes is controlled by transcription factors.

Transcription factors are proteins that bind to DNA regulatory sequences upstream

of the site at which transcription is initiated. Various regulatory pathways control

their activities (see Chapter 6). More than 5% of human genes encode transcription

factors (Tupler et al. 2001). Once activated, transcription factors bind to gene regula-

tory elements and, through interactions with other components of the transcription

machinery, promote access to DNA and facilitate the recruitment of the RNA poly-

merase enzymes to the transcriptional start site.

In eukaryotes, there are three RNA polymerases, namely, RNAP I, II, and III.

RNAP II catalyzes the transcription of protein-coding genes and is responsible for

the synthesis of mRNAs and certain small nuclear RNAs, while the others are re-

sponsible for generating primarily tRNAs (RNAP III) and ribosomal RNAs (RNAP I)

(Allison et al. 1985).

The RNAP II enzyme itself is unable to initiate promoter-dependent transcription

in the absence of complementing factors. It needs to be supplemented by so-called

general transcription factors (GTFs) (Orphanides et al. 1996). RNAP II together with

these GTFs and the DNA template form the pre-initiation complex, and the assem-

bly of this complex is nucleated by binding of TBP (a component of TFIID) to the

“TATA box” (Woychik and Hampsey 2002). The TATA box is a core promoter (or

minimal promoter) that directs transcriptional initiation at a short distance (about

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8 Modeling of Gene Expression

30 bp downstream). Soon after RNAP II initiates transcription, the nascent RNA is

modified by the addition of a cap structure at its 5' end. This cap serves initially to

protect the new transcript from attack by nucleases and later serves as a binding site

for proteins involved in export of the mature mRNA into the cytoplasm and its trans-

lation into protein.

The start of RNA synthesis catalyzed by RNAP II is the transcription initiation.

During transcription elongation, the polymerase moves along the gene sequence

from the 5' to the 3' end and extends the transcript. The transition between these

early transcriptional events, initiation and elongation, seems to be coordinated by

the capping process. A family of elongation factors then regulates the elongation

phase (Uptain et al. 1997). Upon reaching the end of a gene, RNAP II stops tran-

scription (termination), the newly RNA is cleaved (cleavage), and a polyadenosine

(poly(A)) tail is added to the 3' end of the transcript (polyadenylation). The resulting

pre-mRNA contains coding sequences in the gene (exons) that are divided by long

non-coding sequences (introns). These introns are removed by pre-mRNA splicing.

Transcription, i.e., the transfer of information from DNA to RNA, and translation,

i.e., the transfer of information from RNA to protein, are spatially separated in eu-

karyotes by the nuclear membrane; transcription occurs in the nucleus, whereas

translation is a cytoplasmic event. For this reason, processed mRNAs must be trans-

ported from the nucleus to the cytoplasm before translation can occur. The bidirec-

tional transport of macromolecules between nucleus and cytoplasm occurs through

protein-covered pores in the nuclear membrane. The export of mRNA is mediated

by factors that interact with proteins of the nuclear pores and bind to mRNA mole-

cules in the nucleus and direct them into the cytoplasm. Translation of mRNA into

protein takes place on ribosomes, i.e., large ribonucleoprotein complexes, and fol-

lows principles similar to those of transcription. Important for the translation pro-

cess is the presence of transfer RNA molecules (tRNAs), which deliver the correct

amino acid to the currently considered nucleotide triplet. tRNAs have a common

characteristic secondary structure and are bound to the mRNA by means of anti-

codons complementary to the triplet for which they carry the appropriate amino

acid. Subsequently, tRNAs are recruited and the polypeptide is synthesized until the

first stop codon is present. The first step is the location of the start codon in conjunc-

tion with subunits of the ribosome triggered by translational initiation factors. Sub-

sequent phases are elongation and termination. The nascent polypeptide chain then

undergoes folding and often post-translational chemical modifications to generate

the final active protein (cf. Section 2.4).

8.2

Promoter Identification

Transcriptional control is very important in gene expression, which makes biological

sense since the cell invests energy to synthesize products and this energy should not

be wasted through subsequent termination of the activity of these products. Gene

transcription is controlled by RNAP II and it depends on the presence of several ad-

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8.2 Promoter Identification