Download Special topics in electrical and systems engineering: Systems Biology

ESE 680-003
Special topics in electrical and systems engineering:
Systems Biology
Pappas Kumar Rubin Julius Halász
Kinetic models:
Lac system
Lac system
mRNA
β-gal
perm
repressor
AlloLactose
Lactose
External
Lactose
Lac system
• Lac operon (three genes)
– Permease, brings in lactose
– β-galactosidase, converts lactose
• Allolactose acts as inducer to the operon
– Binds lac repressor 2:1
Lac system
• Transcription rate:
– Time delay
– Hill
– Basal rate
  M
1  K1 (e
A M )
 dM 
 M
 0


  M
2
K  K1 (e
A M )
 dt transcription
2
Lac system
• Degradation terms
–
–
–
–
apply to all substances
actual degradation
dilution due to growth
effective degradation rate
 dM 
~


(



)
M





M
MM
 dt  decay
Lac system
• Translation
– another time delay
– transcription rate different for each product
 dB 
  B


e
M B
 
B
 dt translation
 dP 
  ( B  P )


e
M  B  P
 
P
 dt translation
Lac system
• Transport terms
– proportional to permease
– bi-directional
Le
 dL 
   LP
K Le  Le
 dt in
L
 dL 
    L P
K Le  L
 dt  out
Lac system
• Conversion terms
– proportional to β-galactosidase
– lactose to allolactose
– allolactose to glucose
L
 dA 
 dL 
  
  AB
 
KL  L
 dt  conversion
 dt conversion
A
 dA 
    A B
KA  A
 dt  gluc
Lac system
• All together now:
Lac system
• Steady state analysis
– Set all equations of motion to zero
• Equilibrium conditions
• Necessary conditions
– Solve one by one
• Arrive at fifth order equation – hard
– Trick: solve for Le=f(A)
Lac system
• Steady state curve has characteristic S-shape
Lac system
• S-shape ensures:
– Hysteresis
– Discrete switching
– Induction
A
L1
L2
Le
Lac system
• Switching and memory
– Need to clear L2 in order to switch up
A
A
t
Le
t
L1
L2
Le
Lac system
• Hysteresis
A
Le
L1
L2
Lac system
• Hysteresis in individual cells
Lac system
Pout
• S-shaped steady state
structure results from
positive feedback
Pin
Aequilibrium
B
P
Te
Le
Lac system
• Switching property is robust
– Model parameters perturbed by 5%
What next?
• Positive feedback leading to bistability is
one of several motifs
• Some control theoretic underpinning
• Many existing models
– Cell cycle
• Challenge is in building the models
– Little parameter information
• Room for theory
Networks and motifs
• Transcription networks
Environment
Signal 1
Transcription
factors
Signal 2
X1
X2
Signal 3
Signal N
X3
XM
Genes
Gene 1
Gene 2
Gene 3
Gene 4
Gene 5
Gene 6
Gene K
Transcription networks
• The lac system has only one (or two) links
Schematic network of transcriptional interactions between group 2 sigma genes
transcription in Synechocystis: The thickness of the arrows is proportional to the effect
of a given mutation on the transcription of the sigma gene to which the arrow points.
Lemeille et al.BMC Microbiology 2005 5:18 doi:10.1186/1471-2180-5-18
Transcription network involving two-component systems. Black single-lined
arrows and T-formed lines show positive and negative transcription regulation,
respectively. Green arrows indicate environmental signal inputs. Double-lined
arrows depict the synthesis of the gene products from the two-component
regulatory genes. The red and blue letters represent RRs and the genes
induced under anaerobic conditions, respectively. For simplicity not all the target
genes for each two-component system are shown, nor the interaction between
ResE-ResD and PhoR-PhoP (13).