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ESE 680-003 Special topics in electrical and systems engineering:
Systems BiologyPappas Kumar Rubin Julius Halász
Kinetic models:Lac system
Lac system
mRNA
β-gal perm
ExternalLactose
LactoseAllo-
Lactose
repressor
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
021
21
)(
)(1
M
M
M
M
AeKK
AeK
dt
dMM
iontranscript
Lac system
• Degradation terms– apply to all substances– actual degradation– dilution due to growth– effective degradation rate
MMdt
dMMM
decay
~)(
Lac system
• Translation– another time delay– transcription rate different for each product
PB
PB
B
B
Medt
dP
Medt
dB
Pntranslatio
Bntranslatio
)(
Lac system
• Transport terms– proportional to permease– bi-directional
LK
LP
dt
dL
LK
LP
dt
dL
e
e
LL
out
eL
eL
in
Lac system
• Conversion terms– proportional to β-galactosidase– lactose to allolactose– allolactose to glucose
AK
AB
dt
dA
LK
LB
dt
dL
dt
dA
AA
gluc
LA
conversionconversion
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
Le
A
L1 L2
Lac system
• Switching and memory– Need to clear L2 in order to switch up
Le
A
L1 L2
A
Le
t
t
Lac system
• Hysteresis
Le
A
L1 L2
Lac system
• Hysteresis in individual cells
Lac system
• S-shaped steady state structure results from positive feedback
Pin
Pout
Le
Aequilibrium
Te
B P
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
Signal 1 Signal 3Signal 2 Signal N
X1 X2 X3 XM
Gene 1 Gene 3 Gene 4Gene 2 Gene KGene 5 Gene 6
Environment
Transcriptionfactors
Genes
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).
