10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 15: Gene Expression and Trafficking Dynamics
This lecture covers: Approach to steady state and receptor trafficking
Central dogma of molecular biology:
DNA
Æ
mRNA
Æ
protein
transcription
translation
Material balance on one specific mRNA
Accumulation = synthesis – degradation
CmRNA ≡
cell volume
mol mRNA
Kr ≡
Vi ≡
moles mRNA
( time )( cell volume )
, transcription (function of gene dosage, inducers, etc.)
cell volume
vessel volume
d ( CmRNA Vi )
= K r Vi − γ r CmRNA Vi
dt
γ r ≡ first order rate constant for mRNA degredation
Vi ≡ a function of time (cells grow, divide)
Æ can’t pull out of the derivative
Do the chain rule:
d Vi
dC
+ Vi mRNA = K r Vi − γ r CmRNA Vi
dt
dt
dCmRNA
1 d Vi
= K r − γ r CmRNA − CmRNA
dt
Vi dt
CmRNA
simplify:
1 d Vi
=μ
Vi dt
(specific growth rate in exponential growth)
dCmRNA
= K r − γ r CmRNA − μ CmRNA
dt
dilution by growth term
(b/c concentration is on a per-cell volume basis)
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
[DD Month YYYY].
dCmRNA
= K r − (γ r + μ )CmRNA
dt
at steady-state:
CmRNA, SS =
Kr
(γ r + μ )
transient case, analytical solution (just integrate)
Kr ⎛
− ( μ +γ r ) t ⎞
⎜1 − e
⎟
(γ r + μ ) ⎝
⎠
CmRNA =
independent of the transcription rate constant K r
S.S.
CmRNA
1
γr + μ
t
Figure 1. Concentration of CmRNA versus time. At long times steady state is
approached.
Similar rate expression for the protein:
(again, per-cell volume basis, analogous constants)
dC p
dt
= K p CmRNA − (γ p + μ )C p
function of time, solved for above
dC p
dt
= Kp
Kr
(1 − e−(γ r + μ )t ) − (γ p + μ )C p
(γ r + μ )
d
=0, t →∞
dt
Kr K p
steady-state:
C p , SS =
(γ r + μ )(γ p + μ )
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 15
Page 2 of 4
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
[DD Month YYYY].
C p , SS
Kp
=
CmRNA, SS
Note:
γp +μ
K p , γ p vary from protein to protein and condition
to condition
Integrate
dC p
:
dt
⎛ (γ r + μ )e − (γ p + μ )t − (γ p + μ )e − (γ r + μ )t
C p = C p , SS ⎜ 1 +
⎜
γ p −γr
⎝
Usually,
in E. coli
γp
γr
ln 2
γr
∼ 7 minutes on average.
for most proteins,
also,
γr
⎞
⎟
⎟
⎠
ln 2
γp
∼ hours to days.
μ
Apply assumptions to get:
Cp =
(1 − e
+ μ)
K p Kr
γ r (γ p
− (γ p + μ )t
)
Delays in synthesis
mRNA – 1 kb gene
Protein – 400 a.a.
E. coli
10-20
20
time (seconds)
Yeast
30-50
20
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Mammals
30-50
60-400
Lecture 15
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Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
[DD Month YYYY].
Cp
t
Delay is generally small compared to
1
γp +μ
Figure 2. Concentration of protein versus time.
However, the delay can dramatically destabilize feedback loops.
Cellular compartmentalization
C p, 1 → C p,2
rate =
where
C p, 1 ≡ C p for compartment 1, and C p, 2 ≡ C p for compartment 2
K transport C p,1
+
kon
koff
o
c y ut.
to
.
prod.
rec.
endocytosis
cell
Figure 3. Diagram of protein-ligand binding on the cell surface.
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 15
Page 4 of 4
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on
[DD Month YYYY].