Download Dynamics of the trp Operon

AKH9
Dynamics of the trp Operon
Refs: Sántilan & Mackey, PNAS 98 (4), 1364-69 [2001]
I. Rahmim doctoral thesis, Columbia Univ., 1990
Alberts, MBOTC III, pp. 417-419, Lewin, Genes V, 435-7
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(Santillan)
Organization of the trp Operon*
• 5 genes code for enzymes:
* Cluster of genes controlled by a single (?) feedback regulatory
mechanism.
The five genes are transcribed as a single
mRNA molecule, allowing their expression to
be controlled coordinately.
There is one promoter.
Within the promoter is an operator.
Tryptophan repressor can bind to operator and
deny access to RNA polymerase.
Because repressor binding stops transcription the
regulation mode is called negative control.
Repressor Terminology
Activation (binding) of repressor requires
binding of two tryptophan molecules:
Multiple RNA polymerases act together on
an operon; multiple ribosomes act together
on even partially formed mRNAs:
Attenuators…
act to terminate transcription prematurely.
They act before or within the coding region
When they are non-functional, transcription
continues past attenuation region
When they are functional, termination occurs
Tryptophan synthesis: chorismate
l-tryptophan
(the 5 enzymes that make tryptophan)
trp E
trp D
trp C
trp B
trp A
Anthranilate synthase
Phosphoribosyl anthranilae
transferase
Phosphoribosyl
isomerase/indoleglycerol
phosphate synthase
Tryptophan synthetase a
Tryptophan synthase b
Tryptophan Operon Promoters:
p1 (strong): Active repressor binds to trp operator and
overlaps p1 with physical exclusion of RNA polymerase
p2 (low efficiency): Within distal part of trpD, results in
production of mRNAs C, B, and A.
Thus, production of trp E and D is differentiated from
production of trp C, B, and A.
Additionally, deactivation and degradation of mRNA
CBA is much slower than for mRNA ED.
Just before trpE transcription is regulated through
attenuation. At low tryptophan levels, reduces by 8 
Enzymatic Control of Tryptophan Levels:
Production: Anthranilate synthase inhibited up to 100% by
tryptophan.
Interconversion: Tryptophanase (tna) catalyzes tryptophanindole interconversion.
Transport Control of Tryptophan levels:
aro P product facilitates transport of aromatic amino acids.
mtr product facilitates tryptophan transport. (mtr is activated
with phenylalanine present, tryptophan absent.)
Product of tyrR modulates expression of aroP and mtr.
Transport dynamics have not been studied – we don’t know the
transporter kinetic coefficients.
Goodwin’s compartmental triad in bacterial
gene expression:
d [mRNA]
 kmRNA  r  [ gene]  kd ,mRNA  [mRNA],
dt
d [enzyme]
 kenzyme  [mRNA]  kd ,enzyme  [enzyme],
dt
d [ product ]
 k product  [enzyme]  kd , product  [ product ]
dt
r (efficiency of feedback repression by product)
=
1
 product
1 
 K D , product



n
(a negative-going Hill coefficient)
Hill Functions: F, F+, F1.00
0.75
Y 0.50
0.25
0.00
0.00
x
, =1
 x
x
4
 x
4
1.00
2.00
X
4
, =1

4
 x
4
4
, K =1
All of these can be written with the non-dimensional variable, x / 
Overall Model for trp Operon
About concentrations:
• Molecules per cell (with a 'standard' cell
size). Intuitive but not usually measured.
• Standard chemical concentrations, e.g.
moles/ml, etc. Normal measurement.
• Stochasticity issues. 50 molecules per cell
of volume 1m3 ~ 0.1 M.
The Effect of Cell Expansion on
Concentration
• Expanding (dividing) cultures lose
concentration even without degradation –
• amount in one cell is unaffected until
division, and is then suddenly halved.
• concentration decreases steadily up to,
during, and after division.
• Intracellular heterogeneous kinetics are
affected by concentration, not amount.
Allowing for Volume Expansion
Without Expansion:
dc generation 

 Vkd c

dt  rate

With expansion:
V
dc
dV generation 
+c

 Vkd c

dt
dt  rate

which can be written:
V
dc generation 
dV generation 
d lnV 

V

 Vkd c  c

 V  kd 
c



dt  rate
dt  rate
dt 



d lnV
The term
is frequently designated as . Expansion
dt
increases the apparent degradation rate. It can be the most
significant factor affecting metabolite concentrion during growth!
Cell Volume
Vc (in liters) = 4 10
-15 60 
e
-1
where  is the specific growth rate, min .
mRNA Equations
d [mRNA ED]
 v p1 r a  gT
dt
 (kd ,mRNA ED   )[mRNA ED]
d [mRNA CBA]
 v p1 r a  gT  v p 2  gT
dt
 (kd ,mRNA CBA   )[mRNA CBA]
Repression:
[apoR]  [tryp]
apoR  tryp, K r1 
apoR+ tryp
[apoR  tryp]
[apoR  tryp]  [tryp]
holo R, K r1 
apoR  tryp  tryp
[holo R]
[holo R]  [operator]
bound repressor, K r2 
holoR  free operator
[bound repressor]
"free" + "occupied" operators = g T ;
apoR + apoR  tryp + holoR + bound repressors = r0 ;
"occupied" trp operators = repressors bound to trp operators.
Intracellular Concentration* of Bound
trp Repressors:
B  B  4 gT r0
2

2
where
B   r0  gT  K r 2  (1  A  A2 )  ,


and
K r1
A
[tryp]
__________________________________
* = molecules per cell/ (cell volume  NA)
Finally, for repression:
conc'n of free trp operators
r 
gT
conc'n of occupied trp operators
 1
gT
conc'n of bound trp repressors
 1
gT
Attenuation
trp leader region: Transcription is
stalled on the pause site until a
ribosome arrives at base 27
How long does RNA polymerase take to move
from the pause site to the potential termination
site, i.e. what is T?
How far does the ribosome travel during time T?
Simulation:
RNA polymerase transcribes at an average rate
of 42 nt/sec, but follows Poisson distribution.
Ribosome adds residues at a corresponding
average rate of 14/sec, but dependent on amino
acid availability. Rate is maximum at high, and
zero at zero, concentration.
vribosome
[ A]
 14 
(residues/sec.)
K a  [ A]
Assume tryptophan concentration controlling.
Monte Carlo Simulation Results:
a
3.0  [tryp] / K a
0.12 
0.4  [tryp] / K a
Other Coefficients Needed
• mRNA Degradation Rates (t1/2 values)
• Transcription Initiation Rates
Enzyme Balances
dz3
 vz 3[mRNA ED]  (kd , protein   ) z3
dt
dz4
 vz 4 [mRNA CBA]  (kd , protein   ) z4
dt
(kd , protein   ) z3
(kd , protein   ) z4
vz 3 
; vz 4 
[mRNA ED]
[mRNA CBA]
Biosynthesis Pathway
K I , ASase
d [anth]
 Vmax, ASase 
 [ ASase]
dt
K I , ASase  [tryp]
[anth]
Vmax, PRTase 
 [ PRTase]    [anth]
K PRTase  [anth]
and similarly for CdRP and tryp (with the addition of the
utilization term, - U c , in the tryp equation. The term in red
represents inhibition of the first enzyme by tryp.
Redux: Overall Model for trp
Operon