Constrained fuzzy logic. (A) Constrained fuzzy logic describes interactions between biological species with logic gates. The logic gates are evaluated based on the output of the transfer function (f) that quantitatively relates the input and output species. In this example, AND gates are evaluated with the PRODUCT operator and OR gates are evaluated with the SUM operator. Evaluation of the AND and OR gates with the MIN and MAX operators, respectively, is also supported by Q2LM. Note that the SUM operator is not identical to arithmetic sum, but rather, the logical sum of two possible values is equal to the first plus the second minus the product of the two (i.e., V₁ + V₂ – V₁V₂, where V₁ is the value of one possible output and V₂ is the value of the other). (B) The quantitative relationship between any two species is specified with a transfer function. In this paper, we use a normalized Hill function multiplied by a gain as the transfer function, although other functional forms can easily be imagined.

Converting posited interactions of intracellular signaling into a logic model. (A) The relationship between species in an intracellular signaling network is depicted graphically. Grey dashed blunted arrows indicate inhibitory interactions. (B) To convert the posited interactions in A into a logic model, we consider if the logic describing the relationship between input and output species should include an AND gate for species with more than one input, and find that AND gates are necessary for description of formation of AP1 homo- and heterodimers. AND gates are indicated by the input species linked to a small circle, which is further linked to the output species. (C) The logic model is recorded as a spreadsheet to be loaded into the Q2LM software. The first three columns specify which species interact as well as the logic of these relationships. The last three columns specify the parameters of the transfer functions of the interaction contained in that row. (D) Q2LM has been specifically designed to ask academically and industrially relevant questions.

Q2LM files for examining intracellular signaling logic model. (A) Example of a scenario file that Q2LM imports to simulate experimental perturbations in a variety of environmental conditions. A detailed description of all file types is provided in the software's manual. In this case, environments with partial or full stimulation of TNF-α and TGF-α alone or in combination will be simulated with inhibition of the “Experimental” species JNK, ERK, and MEK at levels listed in the “Values” column alone or in combination, where the maximum number of species to inhibit at any one time is listed in the “MaxNum” column. (B) Example of a criteria file. Simulation results from environments with perturbation are compared to environments without perturbation and Q2LM calculates if the criteria have been met. In this case, the criterion is that the AP1HomoDim species increase in value by at least 0.25 with perturbation compared to without. (C) Example of portion of a Results file Q2LM outputs to indicate, for each environment, the values of perturbation that met the criteria in 3B and in what fraction of models they were effective. Ellipsis indicates conditions of intermediate doses that were not included. Tested environments for which no perturbation met the criteria are not listed.

Species values as a function of simulation step for intracellular signaling model. (A) For each indicated species, the median value for all models at the final 19 simulation steps is shown (Q2LM does not save all simulation steps when memory is a limitation) along with the final value calculated by the solver, which has been copied several times for visualization. Upper and lower error bars indicate the third and first quartile, respectively. Simulation conditions: TGF-α = 1; TNF-α = 1; Perturbation with different combinations of JNK, MEK, and ERK inhibition is indicated by different line color. Different line styles represent different models. (B) Median value for AP1 homo- or heterodimers with no inhibitor perturbations. (C) Median value for AP1 homo- or heterodimers for inhibitor combinations that met criteria of increasing the value of AP1HomDimer by at least 0.25 in at least 25% of the models.

Development of logic model of G-CSF administration. (A) Depiction of G-CSF pharmacokinetics at the tissue, cellular, and molecular level, adapted from [23]. (B) Logic model based on 5A. All transfer functions have default parameters g = 1; n = 3; and EC₅₀ = 0.5. Arrow labels indicate the following steps of the pharmacokinetics of the molecule: (1) When G-CSF is administered intravenously (doseGCSF), it enters the bloodstream where it is subject to (2) nonspecific clearance (clearance). (3) Precursor neutrophils (pN) possess receptors (pNR), which (4) bind G-CSF in the blood (pNboundGCSF). (5) Bound G-CSF can be degraded (pNdegGCSF), and (6) what is not degraded is recycled back into the bloodstream (pNrecGCSF). (7) Bound G-CSF also stimulates proliferation and differentiation into mature neutrophils (N). (8) Mature neutrophils possess receptors (NR) that can (9) bind G-CSF (NboundGCSF). Bound G-CSF is then (10) degraded (NdegGCSF) or (11) recycled (NrecGCSF). (12) Value of G-CSF in the blood (bloodGCSF) is limited by the dose, clearance, and amount recycled. (13) An additional species bodyGCSF represents the exchange of G-CSF from the blood to the body cavity and is necessary in the logic model to ensure that the bloodGCSF node is also limited by its own value. (C) The G-CSF logic model was simulated under non-limiting precursor neutrophils and dose conditions (pN = 1 and doseGCSF = 1) with multiple levels of clearance (0, 0.1, 0.2, etc.). Median value of the neutrophil and G-CSF levels in the blood nodes (N and bloodGCSF) were plotted as a function of simulation step, with error bars indicating the first and third quartile of predictions of 100 models with normally distributed noise with a standard deviation of 5% added to the transfer function parameters. As levels of clearance decreased, maximal values of N and bloodGCSF increased as well as the number of simulation steps until the species values decreased to zero. Further analysis indicated adding noise with a standard deviation of up to 25% led to identical conclusions for all results.

Effect of perturbations to G-CSF pharmacokinetics on criteria. In all parts, perturbations to (A, B) species or (C, D) model parameters were made when the G-CSF logic model was simulated under non-limiting precursor neutrophils and dose conditions (i.e., pN = 1 and doseGCSF = 1) with multiple levels of clearance (0, 0.1, 0.2, etc.), with each color and line style corresponding to a different fixed value of the clearance species as shown in the legend in the rightmost panel for each part. Median effects are plotted, with error bars indicating the first and third quartile of predictions of 100 models. (A) The median effect of increasing inhibition of the pNdegGCSF and NdegGCSF nodes on each criteria. (B) The median effect of varying the minimal possible value of the pNboundGCSF node. Because the N species was not observed to decay in these simulation, the first panel is the increase in logic steady state value of N, instead of number of steps until decay. (C) The median effect of changing the gain of the transfer function relating pNboundGCSF to pNdegGCSF and NboundGCSF to NdegGCSF on each criteria. (D) The effect of changing the EC₅₀ of the bloodGCSF to pNboundGCSF interaction.

In vivoincrease in WBC associated with decreased G-CSF degradation. White blood cell count was higher in mice treated with a mutant G-CSF engineered for decreased degradation via increased dissociation in the endosomal compartment (D113H) than those treated with wildtype G-CSF. Five animals were treated with 5FU to inhibit haematopoiesis 24 h prior to treatment with the colony stimulating factor. “Veh” denotes sham treatment with PBS rather than 5FU. *p < 0.001 versus vehicle-treated controls.