Fig. 5: A simple approximation for digital logic using a sum of Hill terms recapitulates all monotonic logic functions in a single parameter space.
From: Nonlinear delay differential equations and their application to modeling biological network motifs
a A prototypical regulatory network involving logic where X and Y both regulate Z, which must integrate the two signals using some logic before it can in turn activate a downstream reporter R. b Parameter space showing regions where regulation approximately follows 14 of the 16 possible 2-input logic functions depending on the strength of two single-variable Hill regulation terms (ηZ1: regulation of Z by X, ηZ2: regulation of Z by Y). Network logic can be smoothly altered by varying the parameters (ηZ1, ηZ2), with a change of sign in (n1, n2) required to switch quadrants. The bottom-left quadrant shows that very weak regulation in both terms leads to an always-off (FALSE) function, weak regulation in one arm only leads to single-input (X, Y) functions, strong regulation in both arms leads to an OR function, and regulation too weak in either arm alone to activate an output but strong enough in sum leads to an AND function. The other three quadrants are related by applying NOT to one or both inputs, with function names related by de Morgan’s law70 NOT(X OR Y) = NOT X AND NOT Y. In particular, X IMPLY Y = NOT(X) OR Y, X NIMPLY Y = X AND NOT(Y), X NOR Y = NOT X AND NOT Y, and X NAND Y = NOT X OR NOT Y. Truth tables for all 16 logic gates are provided in Supplementary Table 1 for reference. The two non-monotonic logic functions, X XOR Y and X XNOR Y, are those 2 of 16 not reproduced directly using this summing approximation. They can be produced by layering, e.g., NAND gates70. c Representative time traces for AND (ηZ1 = ηZ2 = 0.9) and OR (ηZ1 = ηZ2 = 1.8) gates with n1 = n2 = −2, n3 = −20, ηR = ηZ1 + ηZ2. The function \(\mathrm{sgn}\,(n)=+1\) when n > 0, \(\mathrm{sgn}\,(n)=-1\) when n < 0.
