Guided example: lac operon

Let’s revisit the lac operon system, now moving from the conceptual graph we built earlier to a Boolean model. In a previous section, we defined the model scope by selecting relevant biological components and mapping their interactions—including directionality and sign (positive, negative) of the interactions —in the form of a graph. To turn this graph into a Boolean model, we now translate each interaction into Boolean logical rules based on our biological understanding. 

Let’s do this step by step:

We know that the system depends on the availability of glucose and lactose. Therefore we choose them as inputs to our system (the values of input nodes can be set by the user, so we will be able to play out several scenarios that would also be experimentally accessible in the lab). Furthermore we have the Lac repressor protein. In addition we have the  polycistronic mRNA that results from transcribing the Lac operon and the lac enzymes that are translated from it. Overall we would be interested to see under which conditions lactose is metabolised. Therefore our model output could be chosen to be a very abstract model component we call lactose metabolism. The output node should always be at least a proxy for something that can be experimentally assessed.

From interactions to Boolean logic

Each component of our Boolean model can be in one of two states—active (1) or inactive (0). The interactions between components, as defined in our conceptual graph, can be translated into Boolean expressions that capture their biological effects. For reference, the model components we are working with are:

• Inputs: Glucose, Allolactose
  
• Internal nodes: Allolactose, Lac repressor, cAMP, CAP, Polycistronic lac mRNA, Lac enzymes
  
• Output: Lactose metabolism
  

Boolean logic has a direct equivalent in spoken language. We can formalise our biological knowledge of the system, i.e. what we know about the effect that absence and presence of the individual biological entities have on the absence and presence of others, into statements that directly translate to Boolean logic.

We now formalise the biological interactions between these components using Boolean logic:

Lac repressor is active if allolactose is not present:
Lac repressor = NOT(Allolactose)

cAMP is only produced when glucose is absent:
cAMP = NOT(Glucose)

CAP is active when cAMP is present:
CAP = cAMP

The lac operon is transcribed only when the repressor is inactive and CAP is active:  
Polycistronic lac mRNA = NOT(Lac repressor) AND CAP

The mRNA is translated into the corresponding enzymes:
Lac enzymes = Polycistronic lac mRNA

The enzymes are needed for lactose metabolism:
Lactose metabolism = Lac enzyme

Each of these logic statements is directly derived from the known biological effects we represented in our conceptual model. These Boolean rules together form a Boolean network, capturing the causal logic of lac operon regulation (Fig 10).

Again, there is not just one possible model describing this process. We could have chosen the model components differently, depending on which components we consider important or are interested in. 

For a more detailed guide on modelling the lac operon in the Boolean framework refer to this publication: A practical guide to mechanistic systems modelling in biology using a logic-based approach (Niarakis A and Helikar T, 2021).

Now move on to the next page to explore how the model behaviour changes in two different scenarios of sugar availability.