Fuzzy Logic Analysis of Kinase Pathway Crosstalk in TNF/EGF/Insulin-Induced Signaling
Figure 2
As an example, local logic gate construction is illustrated for IRS(S) (IRS phosphorylation at serine 636). (A) Logic-based models use incoming edges to contain activity level of input or regulatory network species (for IRS(S), the inputs were TNF, EGF, and time) with the logic gate at the node that performs the logic operation to update output signal (IRS(S)). (B) A Boolean logic gate for IRS(S) could be represented in terms of the logic statement “(TNF or EGF) and (NOT(time))”, represented here in schematic form where the top shape is an “OR-gate” the circle is a “NOT” operation, and the lower left shape is an “AND-gate”). (C) The truth table for the logic in (B) states the output of IRS(S) (0 for off or 1 for on, in bold) based on the input state. (D) To set up a FL gate, the first step is to assign membership functions (MFs) to the input variables (“TNF”, “EGF”, and “time”). In this example, each input variable has two or three membership functions (“L”, “M”, and “H” representing low, medium, and high states, respectively). An MF relates an input value to that state's degree of membership (DOM). MFs for Fuzzy and Boolean (2 MFs)/discrete multi-state (>2 MFs) logic forms are illustrated with the same state thresholds. (E) The simulations from the Boolean logic gate shown in B–C is compared to experimental data and the Fuzzy logic gates specified in F below (see Figure 5A for the experimental and simulation conditions). The BL gate is not able to model intermediate state for smooth transitions, and simulations of the FL gate better fit the data as compared to the BL gate. (F) To set up a FL gate, the MFs for the inputs and the constant values for the outputs are defined. For simplicity, we use normalized input and output values. Next, logic rules are listed as “if A (the antecedent), then B (the consequent)” using the input and output states as descriptors. Weights between 0 and 1 are assigned to each rule (indicated in parentheses), which is helpful for rules that should have minor influence (e.g. rule 4). The rules for IRS(S) are each graphically listed with the outline of the membership functions specified for that rule's antecedent. Inputs not considered for an antecedent are indicated by a light gray box. The consequent for each rule is indicated by a bar whose height is proportional to the rule weight. We do not depict FL rules in a truth table because a row is not necessarily unique in FL (c.f. (C)). (G–H) Two input scenarios are presented to illustrate FL gate computation (horizontal gray arrows) and defuzzification (vertical gray arrow). The amount of color filled in (yellow for inputs and blue for output) is representative of the DOM (for inputs) or degree of firing (DOF) given the input values (for outputs). The input values are listed on the top and indicated graphically by the vertical red lines. For example in scenario 1, rule 1 fires (full dark blue bar) because the antecedent (TNF is H) has a high DOM (filled in yellow). The firing strength of the rule is the minimum of the antecedents; therefore, rule 2 does not fire because while time has low DOM to L (∼.4) and the DOM of time to H is near zero. To defuzzify (resolve the output value given a set of firing rules), an average is computed from the output values of each rule weighted according to both firing strength and rule weight (see Methods). The bottom row in the consequent column shows the aggregated outputs and the small red line is the defuzzified or final, value. The scenario illustrations were adapted from the “rule viewer” in Matlab's Fuzzy Logic Toolbox.