Genome Informatics 16(1): 73–82 (2005)
73
On the Kinetic Design of Transcription
Thomas Höfer
Malte J. Rasch
[email protected]
[email protected]
Dept. of Theoretical Biophysics, Institute of Biology, Humboldt University Berlin
Invalidenstrasse 42, 10115 Berlin, Germany
Abstract
We analyse a stochastic model of transcription that describes transcription initiation by promoter activation and subsequent polymerase recruitment. Explicit expressions are derived for the
control of an activator on the mean mRNA number and for the mRNA noise. Both properties
are strongly influenced by the kinetics of promoter activation, mRNA synthesis and degradation.
Low transcriptional noise is obtained either when the transcription initiation complex has a long
life-time or when its components associate and dissociate rapidly. However, the ability of an activator to regulate the mRNA level is low in the first and high in the second case. Large noise is
generated when the initial activation step of the promoter is slow. In this case, transcription can
be burst-like; the mRNA distribution becomes bimodal while regulability of the mean copy number
is maintained.
Keywords: gene expression, stochastic dynamics, Markov process, rapid-equilibrium approximation
1
Introduction
Gene expression is tightly controlled by a complex machinery that ensures that a particular protein is
synthesized only when required [4, 24]. At the same time, gene expression may be subject to stochastic
influences. A typical protein-coding gene is transcribed from a small number of alleles, typically one
or two, so that fluctuations in the assembly of transcription complexes on these alleles may lead
to temporal variations in the number of mRNA copies. Such variations could be of considerable
magnitude because the mRNA numbers for many genes appear to be rather small [10].
Stochastic effects in gene expression have therefore become an area of intense theoretical investigation, showing that both transcription and translation may be sources of appreciable noise in mRNA
and protein numbers [8, 13, 19, 28]. More recently, such noise has been characterized in prokaryotes
and yeast by quantitative measurements [3, 22, 25, 26]. In particular, Raser and O’Shea [25] have
found that random fluctuations in protein concentration may behave very differently as the expression
of a gene becomes activated. For some genes, the relative standard deviation of the noise is inversely
proportional to the square root of the protein concentration. This case corresponds to the typical law
of large numbers found for the fluctuations in a great variety of systems [29]. Such a behavior occurs,
for example, when the protein number in the population obeys a Poisson distribution. However, there
are also genes for which the noise is much larger than would be expected from a Poisson distribution.
They reason that the differences in fluctuations of various genes may be due to differential kinetics of
the assembly of the transcription complexes at their promoters. Moreover, for many genes bimodal
expression patterns have been observed, and it has been hypothesized that they result, at least in
part, from stochastic effects in transcription [12, 16, 17, 30].
Here we present a stochastic model of gene transcription that accounts, in a simple manner, for
multi-step process of transcription initiation. Analytic expressions are derived for how the number
of mRNA and its fluctuations depend on the kinetic parameters of transcription complex assembly,
74
Höfer and Rasch
Figure 1: Model of transcription. Transcription can proceed when the promoter becomes activated,
enabling the recruitment of mRNA polymerase and transcription start. For further explanation see
text.
mRNA synthesis and degradation. The model shows that the relative rates of these processes determine both the extent of mRNA fluctuations and the ability of transcriptional regulators (activators
or repressors) to control mRNA number. In particular, two qualitatively different response patterns
to regulators are obtained: a unimodal mRNA distribution shifting smoothly upon changes of regulators and a bimodal mRNA distribution where regulators control the fraction of non-expressing and
expressing cells. These patterns may underlie the experimentally described graded and binary modes
of gene induction [17].
2
Stochastic Model of Transcription
The transcription of most eukaryotic genes requires the binding of gene-specific regulators which then
recruit other proteins, including chromatin-modifying enzymes, co-transcription factors, the general
transcription factors and RNA polymerase II (RNAP) [4]. A greatly simplified scheme of this process is
shown in Figure 1. The multitude of steps leading to an active transcription complex are represented
only by two transitions. First, the unoccupied promoter, to which RNAP is unable to bind (state
1), is converted into an active promoter, to which RNAP can be recruited (state 2). Second, RNAP
binding leads to the preinitiation complex, from which transcription can start (state 3). Both promoter
transitions are reversible, being characterized by forward and backward rate constants α1,2 and β1,2 ,
respectively. In particular, α1 is taken proportional to the concentration of a transcriptional activator
that is assumed to bind during this step. When starting RNA synthesis RNAP leaves promoter, which
now becomes available for a further round of RNAP binding and transcription initiation. Transcription
start from the pre-initiation complex occurs with rate θ, and mRNA is degraded with rate constant
δ. The model incorporates two fundamental properties of transcription: A promoter must first be
activated by binding of transcription factors and, in many cases, by chromatin remodeling to become
competent for binding RNAP. Once this has been achieved, multiple rounds of transcription initiation
are possible [4].
Let pi (m) be the probability that the promoter is found in state i and m mRNA copies are
present. The evolution of pi (m) is described by a system of master equations [29], which for the
scheme of Figure 1, pi (m) take the form
d
p1 (m) = −α1 p1 (m) + β1 p2 (m) + δφ(p1 (m))
dt
d
p2 (m) = α1 p1 (m) − (β1 + α2 ) p2 (m) + β2 p3 (m) + θp3 (m − 1) + δφ(p2 (m))
dt
d
p3 (m) = α2 p2 (m) − β2 p3 (m) − θp3 (m) + δφ(p3 (m)).
dt
(1)
(2)
(3)
Kinetic Model of Transcription
75
The function φ describes the mRNA degradation term in each equation
φ(pi (m)) = (m + 1)pi (m + 1) − mpi (m).
The system is defined for all m ≥ 0 where it is understood that p3 (−1) ≡ 0. Neglecting the delay
between initiation of RNA synthesis and the appearance of the final mRNA product in the θp3 (m − 1)
term in Eq. (2) is justified since the following analysis will be confined to the steady state.
The probability to find m mRNA copies transcribed from one allele is given by
PI (m) =
3
X
pi (m).
(4)
mk pi (m)
(5)
i=1
Let
(k)
µi
=
∞
X
m=0
denote the kth moment of pi (m). Then we obtain for the mean and variance of the single-allele mRNA
distribution PI (m)
m=
3
X
(1)
σ2 =
µi ,
i=1
3
X
i=1
(2)
µi − m2 .
(6)
Typically, two alleles are present for protein-coding genes. When they are characterized by identical
parameters and transcribed independently of one another, the probability to find in total m mRNA
copies can be calculated from the single-allele distribution as:
PII (m) =
m
X
k=0
PI (k)PI (m − k).
(7)
(k)
The moments µi can be obtained explicitly from the master-equation system (1)–(3). For the
following analysis, we consider the steady state of transcriptional activity. Setting the time derivatives
in Eqs (1)–(3) to zero, one finds, after some algebra, that the stationary moments are obtained by
solving a hierarchy of linear equations for ascending k ≥ 0:

³

−(α1 + kδ)
β1
0

 (k)
α1
−(β1 + α2 + kδ)
β2 + θ
= b(k) ,

µ
0
α2
−(β2 + θ + kδ)
(k)
(k)
´
(k) T
where µ(k) = µ1 , µ2 , µ3
. The inhomogeneities read
b(0) = 0,
b(1)
and, for k ≥ 2,
b(k) = −δ
(8)
k−2
X
j=0
(−1)k−j
à !
k
j


0


=  −θµ(0)

3
0




µ(j+1) −  θ
Pk−1
j=0
(9)
0 !
Ã
k
j
0
(j)
µ3



.

(10)
76
3
3.1
Höfer and Rasch
Results
Average Number and Fluctuations of mRNA Copies
Using the solution procedure outlined in the preceding paragraph, one finds for the average number
of mRNA copies at steady state
θ/δ
µ
¶
m=
.
(11)
β
θ
+
β
1
2
1 + 1 + α1
α2
The maximal copy number is mmax = θ/δ. Achieving this will require, in particular, that the second
forward transition of the promoter becomes very fast: α2 À θ + β2 .
It is instructive to quantify how transcription is influenced by the activator concentration. The
relative change of the mRNA concentration versus the relative change in α1 is
ρ=
µ
α1 α2 + β2 + θ
∂ ln m
= 1+
∂ ln α1
β1
β2 + θ
¶−1
.
(12)
This definition corresponds to the concentration control coefficient of metabolic control analysis [9]
and provides a measure for the regulability of transcription by the activator. Specifically, ρ decreases
with the rate constant of the second forward step, α2 . Thus we conclude that the recruitment of
RNAP to the promoter, as measured by α2 , must not be too efficient if the preceding binding of the
activator is to exert significant control on transcription rate.
The following result is obtained for the variance of the mRNA distribution


σ2 = m 
1 +


θ/δ
¶
− m
.
θ + β2 + δ
1
1 + 1 + α β+
α2
δ
1
µ
(13)
The mRNA fluctuations are best characterized by their relative size, which we will refer to as the
noise,
s
σ
f
=
.
(14)
ν=
m
m
In the second of these expression, we have rewritten the noise as a function of the so-called noise
strength (or Fano factor) f = σ 2 /m. In terms of the kinetic parameters, the noise strength f equals
the expression in the square brackets in Eq. (13).
3.2
Kinetic Design of Transcription
When analysing Eqs (11), (12) and (13), one of the kinetic parameters can be fixed by the scale of
time measurement. Without loss of generality, we set δ = 1, so that all other parameters are now
taken relative to the mRNA degradation rate.
The kinetics of transcription are discussed from different points of view. It is generally presumed
that transcription is mediated by the formation of stable complexes between transcription factors and
DNA [24]. This may occur by high-affinity binding, so that any component in the forming transcription
complex on DNA has a long life-time. This case will be referred to as irreversible complex formation.
As second possibility is that individual components associate and dissociate rapidly, while the complex
as a whole remains stable. We refer to this situation as rapid exchange of components. As a third case,
we consider the situation that only some steps in transcription initiation are rapid while others are
comparatively slow. In particular, the remodeling of chromatin structure may require several hours,
after which repeated transcription initiation can take place [18]. We will consider this as a third case
Kinetic Model of Transcription
77
of slow promoter activation. These three kinetic designs turn out to have very different properties
with respect the control of mRNA synthesis and noise.
Irreversible complex formation. Stable complexes between transcription factors and DNA occur
when the dissociation constants β1 and β2 are very small. In the limit β1,2 → 0, one finds
m=
θ
1 + θ/α2
and 0.75 < f < 1.
(15)
Thus transcription can proceed and, depending on α2 can be close to its maximal rate θ. Detailed
analysis of Eq. (13) shows that the noise is comparatively small; the noise strength can become even
somewhat lower than for a Poissonian. However, these properties are achieved at the expense of
the control of the activator on transcription becoming very small. Indeed, in the limit of vanishing
dissociation rate constants β1,2 we have ρ = 0. This expresses the fact that even low amounts of
activator will suffice to fully induce transcription. Thus, despite the high fidelity of transcription, this
design may not be advantageous for genes that need to tightly regulated.
Rapid exchange of component. This is modeled by setting both association and dissociation rates
to comparatively large values. Then individual proteins reside in the preinitiation complex only for
short times. Assuming α1,2 and β1,2 to be much larger than θ and δ, one obtains approximately
m=
µ
θ
β1
1+ 1+ α
1
¶
β2
α2
and f = 1.
(16)
√
The noise is again comparatively small and follows exactly the inverse square-root law ν = 1/ m.
However, in contrast to the previous case, the activator retains control over transcription as can be
seen by the non-vanishing value of ρ (see Eq. (12)).
An analytic solution for the mRNA distribution can be obtained by exploiting the time-scale
difference between the promoter transitions on the one hand and mRNA synthesis and degradation
on the other. The promoter states will rapidly equilibrate and then fulfill the relations
p1 (m) =
PI (m)
,
D
p2 (m) =
α1 PI (m)
,
β1 D
p3 (m) =
α1 α2 PI (m)
,
β1 β2 D
(17)
where D = 1 + α1 /β1 (1 + α2 /β2 ). Summing Eqs (1) through (3) and using these relations, one finds
a master equation for PI (m) containing only the slow processes:
d
θ [PI (m − 1) − PI (m)]
µ
¶
PI (m) =
+ δ [(m + 1)PI (m + 1) − mPI (m)] .
|
{z
}
dt
β2 1 + β1
1+ α
α1
mRNA degradation
2
|
{z
mRNA synthesis
(18)
}
Eq. (18) describes a birth-and-death process of mRNA with the effective transcription rate
θeff =
θ
µ
β2 1 + β1
1+ α
α1
2
¶,
(19)
which is controlled by the promoter transitions. The stationary solution is the Poisson distribution [29]
PI (m) =
m
θeff
exp{−θeff }.
m!
(20)
The close match between the exact steady-state solution of Eqs (1)–(3) and the Poisson distribution
resulting from Eq. (18) is illustrated in Figure 2A. Figure 2B shows a corresponding sample time
course of mRNA computed with Gillespie’s algorithm [7].
78
Höfer and Rasch
Slow promoter activation. This can be accounted for by decreasing the rate of the first promoter
transition. Such slow reversible processes induced by activator binding can be histone acetylation and
ATP-dependent chromatin remodeling. We have found in numerical simulations that the rate of the
initial step exerts strong control on the size of the mRNA fluctuations. This can be understood as
follows. When both the activation step (state 1 → state 2) and its reversal are relatively infrequent
events, many rounds of transcription initiation will take place once the activated state has been
reached. In this way, slow fluctuations in promoter activation and inactivation will become amplified
by the mRNA synthesis.
Figures 2C and 2E show the mRNA distributions obtained with a progressively slower first step.
Both forward and backward rate constants, α1 and β1 , were decreased simultaneously going from
Figure 2A and B to C and D and finally to E and F, while keeping their ratio constant. As a result,
the mean mRNA number stays constant throughout. In Figure 2C, α1 and β1 are comparable to the
rate of mRNA degradation. The noise is considerably larger as seen both by the spreading of the
distribution (Figure 2C) and the higher amplitudes in the sample time course (Figure 2D). When the
time hierarchy between slow initiation and fast subsequent steps becomes even more pronounced, the
probability distribution changes to a bimodal shape with two representative levels of transcriptional
activity (Figure 2E). The sample time course is characterized by random switching between an offstate and an on-state of transcription (Figure 2F). This burst-like behavior occurs because the slow
initial activation step is followed by repeated round of transcription initiation. Once the promoter
becomes inactive again, it remains so sufficiently long that mRNA can become completely degraded.
The bimodal mRNA distribution can be characterized quantitatively by reducing the master equation (1)–(3) to a two-state system in which the first promoter transition is the rate-limiting step.
P
Accordingly, we define the probability to be in the on-state as Pon = ∞
m=0 [p2 (m) + p3 (m)]. We
can now apply a rapid-equilibrium approximation to the switching between state 2 and 3 in a similar
manner as has been done above for both promoter transitions (cf. Eq. (17)). In this way, we obtain
for the dynamics of Pon
β2 + θ
d
Pon = α1 (1 − Pon ) − β1
Pon .
(21)
dt
α2 + β2 + θ
The mean mRNA number in the on-state is given by
mon =
θ
.
β
θ
1 + 2α+
2
(22)
Because the mRNA number in the off-state is practically zero, we have m = Pon mon , where Pon
is the stationary solution of Eq. (22). As Figure 2F shows, the mRNA fluctuations are dominated
by the switching between off-state and on-state, while the fluctuations in the on-state are small by
comparison. Neglecting the latter, one obtains for the noise strength
f = mon (1 − Pon ) .
(23)
For a bimodal distribution, we have Pon < 1. The mean mRNA number in the on-state can be raised
by increasing θ and α2 . Therefore the noise strength can become arbitrarily large.
3.3
Patterns of Transcription Activation
So far we have considered how well transcription can be regulated by an activator affecting α1 based on
the control coefficient ρ. It is evident from Eq. (12) that the timescale of the first promoter transition
has no effect on ρ but only the ratio α1 /β1 . Therefore, the regulability can be equally good for kinetic
designs with rapid and slow promoter activation. How do the different noise characteristics affect gene
induction?
Kinetic Model of Transcription
79
Figure 2: The promoter kinetics control the mRNA fluctuations. The left column shows stationary
mRNA distributions for different velocities of promoter activation; the right columns gives corresponding sample time courses. A and B: Fast activation (α1 = 101.5 ); the exact solution of Eqs (1)–(3) shown
by the black bars is matched closely by the Poisson distribution (Eq. (20), solid line). C and D: Slower
activation (α1 = 1). E and F: Very slow activation (α1 = 10−1.5 ); the bottom trace in F shows when
the promoter is in the ‘on-state’ (state 2 or 3). Other parameters: β1 = 2.96, α1 , α2 = 100, β2 = 1,
and θ = 50; all parameters are in units of the mRNA degradation rate constant.
To analyse this question, Eqs (1)–(3) were solved numerically for varying values of α1 while holding
the other kinetic parameters constant. For appropriately chosen parameters, the average mRNA concentration increases significantly as a function of α1 (see Eq. (11)). However, this increase can become
manifest at the single cell level in two fundamentally different ways (Figure 3). When the promoter
transitions are all rapid and reversible (rapid exchange of components), the mRNA distribution PI (m)
is centered around the mean. If in such a system mRNA levels in the cells of a population are recorded
for increasing activator concentration, one will observe a continuous shift of the entire cell population
to higher expression levels, as shown in Figure 3A. Figure 3B depicts the distributions at the activator
concentrations indicated in Figure 3A; they are all nearly Poissonian (note the correspondence to
Figure 2A).
A different picture is obtained when the promoter transitions are reversible but the first step is slow
(slow promoter activation). Instead of a homogeneous increase of mRNA in the cell population, some
cells begin to switch to an elevated mRNA level at a critical activator concentration (Figures 3C, D).
Thus a bimodal mRNA distribution emerges (corresponding to the case depicted in Figure 2E). As the
activator concentration increases, more cells will be found in the on-state, while the mRNA number
in the on-state itself remains constant. The distribution for two independent alleles corresponding
80
Höfer and Rasch
Figure 3: Graded and binary responses to a transcriptional activator. The upper row shows contour
plots of the mRNA distribution and the lower row selected distributions at the activator concentrations
indicated. A and B correspond to the case of rapid component exchange shown in Figure 2A. C and D
correspond to the case of slow promoter activation shown in Figure 2E. E and F depict the two-allele
distribution for this case (see Eq. (7)). The insets in B and D visualize the distributions (2) through
depicting a corresponding realization of mRNA numbers in a cell population in each case, coded by
grey level.
to this case is shown in Figures 3E and F. Note that there is a considerable range of intermediate
activator concentrations where only one allele is transcribed.
4
Discussion
Our analysis of the model of transcription has yielded analytic expressions for how mRNA number and
its fluctuations are controlled by the kinetics of promoter activation, mRNA synthesis and degradation.
Compared to previous analytic studies of noise in gene expression [13, 28], we have focussed on a more
detailed description of promoter activation and polymerase recruitment that enabled us to characterize
different kinetic designs of this process.
A central finding of this study is that a relatively simple model of gene transcription can give rise
to quite different kinds of mRNA distribution. The two distinct patterns of gene induction seen in
Figure 3 strikingly resemble the two principal modes of transcriptional regulation – graded and binary
– that have been described in the experimental literature [2, 12, 15, 16, 17, 30]. In both modes the
mean protein expression in a cell population increases with the concentration of an activator. In the
graded, or rheostatic mode, a dose-dependent gradual increase is seen in every cell. This is exemplified
by the model solutions shown in Figures 3A and B. In the binary, or probabilistic mode one observes
only discrete states of gene expression in individual cells, typically an off-state in which little or no
protein is found and an on-state in which the protein occurs with a certain concentration. Thus two
cells experiencing the same stimulus may have fundamentally different protein concentrations. With
increasing activator, more cells will be found in the on-state. This scenario corresponds to the model
solutions depicted in Figure 3C–F.
Kinetic Model of Transcription
81
An important prediction of the kinetic model for binary gene expression is that monitoring expression over long time periods should reveal discrete bursts of transcriptional activity. This has been
difficult to measure so far, but some evidence supporting this prediction has been presented [6, 21].
An important question in this respect relates to the protein dynamics. We have found in an extension
of the model that slow enough protein turnover results in an ‘averaging’ of the protein dynamics over
the mRNA bursts, so that a bimodal mRNA distribution can be converted into a unimodal protein
distribution. Therefore, a kinetic explanation of the binary response mode must also consider the
dynamics of translation and protein degradation.
The kinetic mechanism proposed here may not be the only cause for a binary expression pattern.
Indeed it has been argued that bistability in signal transduction and/or gene regulatory circuits can
give rise to two-peaked protein expression in a cell population [1, 11]. Similarly, very steep thresholds in the transduction of external stimuli yield the same expression pattern ([27] and Baumgrass
et. al. unpublished observations). Interestingly, a recent modeling study has shown that slow kinetics of transcription factor binding and cooperativity may combine in causing binary expression [23].
Further experimental work will be needed to distinguish between these models for particular genes.
For many genes, low expression noise rather than burst-like transcription may be advantageous.
Our analysis has shown that rapid association and dissociation of the protein components of the
transcription complex will favor small noise while retaining regulability of the transcription rates.
Recent experimental studies have provided evidence that such a rapid exchange can occur both for
transcription factors and RNAP [5, 14, 20].
Note: M.R.’s current address is Institute for Theoretical Computer Science Technical University
Graz, Austria.
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