Figure 1.
Modeling techniques balance specificity and complexity. Principal component analysis elucidates correlations among network components (A–E) by a linear transformation of the data, resulting in orthogonal principal components. Bayesian networks use conditional probabilities to associate correlations and influences between network components. Fuzzy logic uses rule-based gates and probabilistic representation of input variables to quantify influences and mechanism that regulate network species. Differential-equations models using mass-action kinetics are highly specified defining regulatory mechanism by defining rates of change in network species concentrations.
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.
Figure 3.
Each subfigure depicts the MFs and logic rules for the FL gates: (A) ERK, (B) MK2, (C) JNK, (D) IKK, (E) MEK, (F) IRS(Y), (G) ProC3, (H) FKHR, (I) Akt, (J) IRS(S), and (K) Casp8. The notation is identical to Figure 2, except that rule weights are specified only when they are not 1 and input and output concentrations are normalized (arbitrary units).
Figure 4.
(A) The original network diagram is adapted from Janes et al. [45] and was used as a starting point to construct the FL gates. Network species whose concentration was measured by Western blot in the data-compendium are notated with a blue square (“pS” for phospho-serine, “pY” for phospho-tyrosine specific antibodies, “clv” for the cleaved form, and “pro” for the uncleaved form). Brown circles mark data compendium proteins measured by kinase assay. (B) This diagram depicts the global FL model, comprised of the 11 local FL gates with time delay and “max” functions. The network topology of the model differs from that of the original diagram.
Figure 5.
The experimental data compendium and simulation of the global FL model.
(A) The left heatmap portrays the averaged normalized data from the experimental compendium [20]. Ten stimulation conditions with TNF, EGF, and insulin (top) are shown with the measurements at 0, 5, 15, 30, 60, 90,120, 240, 480, 720, 960, 1200, and 1440 minutes below. Measurement types (western blot or kinase assay) are indicated in Figure 4A and are described in detail in Gaudet et al. [43]. In the middle, the heatmap shows the results of simulation using the global model under normalized treatment conditions, corresponding with the data compendium shown on the right. Identical simulations of an equivalent discrete logic model (DL, built by changing only the degree of fuzziness from the FL model and leaving the rules and MF thresholds unchanged) are shown on the left (see Methods). The cytokine treatment concentrations are marked directly on the heatmap in ng/mL for the data and arbitrary units for the models. See Figure S5 for an alternative depiction of the data and simulation results. The FL and DL models have fitnesses of 44.6 and 96.7, and normalized fitnesses of 0.035 and 0.076, respectively. (B–D) Simulation and data time courses are plotted for three treatment conditions to highlight cases where the FL model fit the data better than the DL model (B), where both models have similar performance (C), and both models fail (D).
Figure 6.
Model prediction of C225 interference with TNF-stimulated signaling.
A heatmap depicts the experimental and FL-model predicted response of cells co-treated with 5 ng/mL of TNF and 10 µg/mL of C225 (an antibody that interferes with ligand binding to the EGF receptor), as compared to TNF alone. The model fitness without and with C225 are 2.9 and 2.6, respectively.
Figure 7.
(A) A heatmap depicts the data, untrained model (Figure 3B), and trained model time courses for MK2. (B) The regressed rule weights are plotted for the 12 candidate rules. The rules are indicated in tabular format; the first two rows describe the state of the inputs, TNF and time, and the last row is the output MK2 state. L and H represent low and high states, and E is the state describing the early response lag. Symbols above the plot show whether the rules were present (✓) or not applicable () in the untrained model.