The lac operon

The E. coli lac operon controls the production of enzymes needed to metabolize lactose and is regulated in large part by lactose availability. This example model includes

• LacI (I) repression of the lac operon via blocking of the operator (O),

• lactose (L) transport by LacY (Y) permease,

• conversion of lactose to allolactose (A) by LacZ β-galactosidase (Z),

• and allolactose-mediated derepression of the operon.

from pykappa import System

system = System.from_ka(
    """
    %init: 1 O(i[.])
    %init: 1 I(o[.])
    %init: 10 Y()
    %init: 10 Z()

    %obs: 'Extracellular lactose' |L(loc{out})| / 100
    %obs: 'LacY' |Y()|
    %obs: 'Free lac operon' |O(i[.])|

    I(o[.]), O(i[.]) <-> I(o[1]), O(i[1]) @ 1, 0.1   // lac repression
    Y(), L(loc{out}) -> Y(), L(loc{in}) @ 0.01       // lactose transport
    Z(), L(loc{in}) -> Z(), A(z[.], i[.]) @ 1        // lactose to allolactose
    A(i[.]), I(o[.]) <-> A(i[1]), I(o[1]) @ 1, 0.1   // derepression
    O(i[.]), ., . -> O(i[.]), Z(), Y() @ 1           // lac expression
    L(loc{in}) -> . @ 1                              // lactose metabolism

    // degradation
    A() -> . @ 0.1
    Y() -> . @ 0.5
    Z() -> . @ 0.5
    """,
    seed=42
)

The rule graph illustrates the local transformations that the rules specify. The conditions for these transformations are not detailed in the graph as they are in the Kappa string representation.

system.rule_graph()

We simulate the system first without lactose, then add extracellular lactose to observe the regulatory response.

# Simulate with no lactose
while system.time < 300:
    system.update()

# Add extracellular lactose and continue simulating
system.mixture.add("L(loc{out})", 1000)
while system.time < 800:
   system.update()

system.monitor.plot(figsize=(4.5, 4.5));

Without lactose, the operon remains mostly repressed. When extracellular lactose is added,

• existing LacY transports some lactose into the cell,

• intracellular lactose is converted to allolactose by LacZ,

• allolactose inactivates the LacI repressor,

• derepressing the operon and leading to increased LacY and LacZ production.

As lactose is metabolized the operon is again repressed.