ESE Special topics in electrical and systems engineering: Systems Biology Pappas Kumar Rubin Julius Halász Kinetic models: Lac system

mRNA β-galperm External Lactose Allo- 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

Lac system Degradation terms –apply to all substances –actual degradation –dilution due to growth –effective degradation rate

Lac system Translation –another time delay –transcription rate different for each product

Lac system Transport terms –proportional to permease –bi-directional

Lac system Conversion terms –proportional to β-galactosidase –lactose to allolactose –allolactose to glucose

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 L e =f(A)

Lac system Steady state curve has characteristic S-shape

Lac system S-shape ensures: –Hysteresis –Discrete switching –Induction LeLe A L1L1 L2L2

Lac system Switching and memory –Need to clear L 2 in order to switch up LeLe A L1L1 L2L2 A LeLe t t

Lac system Hysteresis LeLe A L1L1 L2L2

Lac system Hysteresis in individual cells

Lac system S-shaped steady state structure results from positive feedback P in P out LeLe A equilibrium TeTe 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 1Signal 3Signal 2 Signal N X1X1 X2X2 X3X3 XMXM Gene 1Gene 3Gene 4Gene 2Gene KGene 5Gene 6 Environment Transcription factors 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 :18 doi: / 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).