Homework 2.2: XOR gates (55 pts)¶
In Chapter 5, we worked out regulatory functions for regulation of a single gene Z by two effectors X and Y based on AND or OR logic. We found that when we considered the concentration of effectors as “inputs” and the resulting concentration of Z as an “output” we could think of the regulatory architecture as a logic gate.
a) We used a formalism to where we wrote down the states of the promoter region (unbound, bound with X, bound with Y, bound with both X and Y) and the respective weights of each state. We then came up with a regulatory function by summing the weights for which the promoter may bind and dividing by the sum of the weights. Now, we’ll work backwards. Write down an expression for a dimensionless regulatory function \(f(x,y)\) that would correspond to an XOR gate. The truth table for an XOR gate is below.
X |
Y |
Z |
|---|---|---|
0 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
0 |
Based on this expression what would have to be true of the promoter architecture in terms of how the effectors may bind? Do you think this scenario would be well-modeled by the regulatory function you wrote?
b) Sketch a genetic circuit that can function as an XOR gate. That is, X and Y are inputs and the output is the concentration of a single gene product Z. You will likely need to have more than just X, Y, and Z in your circuit. Clearly explain the thinking behind the circuit you chose. Are there feedforward loops in your circuit? How about feedback loops?
c) Demonstrate that your proposed circuit works by modeling its dynamics with a system of ODEs. Specifically, plot the response of the circuit to:
A step from \(x = y = 0\) to \(x = y = \text{high}\)
A step from \(x = y = \text{high}\) to \(x = y = 0\)
A step from \(x = y = 0\) to \(x = \text{high}\), \(y = 0\)
A step from \(x = y = 0\) to \(x = 0\), \(y = \text{high}\)