Why is this a powerful approach? Because it can greatly help to understand the system being modeled. We do not wish to engage here in discussing the meaning of 'understanding' and of the usefulness of models; these have been frequent topics in biological discourse in recent years. We will rather illustrate by a hypothetical example, in very broad and practical terms (the interested reader is referred to [8] and references 11, 14, 19, 35, 40, 44 cited therein for detailed descriptions of real world examples and algorithm testing/validation). Suppose we want to investigate how signaling and gene regulation via the epidermal growth factor and tumor necrosis factor-α receptors may be altered in human hepatocarcinoma cells. We would select information available in pathway databases and a relevant transcriptomics and/or proteomics experimental dataset (typically readouts after various perturbations), and then put CellNOptR to work. Some likely results of this exercise in logic modeling could be the predictions that, in contrast to other cells, in these cancer cells the tumor necrosis factor-α receptor does not activate phosphoinositide 3-kinase, both Map3K1 and Map3K7 are required to activate MKK4, an inhibitory link from ERK to SOS-1 may be present, and so on. This context-specific model refinement provides concrete hypotheses: maybe a putative interaction shown in a yeast two-hybrid experiment does not occur *in vivo*, or maybe the transformed cell line phenotype is simply different from the canonical pathway. The latter may prove to be critical information for identifying the mechanisms that cause the hepatocarcinoma cells to respond to stimuli differently than their normal liver cell counterparts. If the experimental data have detailed time-course readouts, the differences obtained when fitting via the different algorithms could lead to additional conclusions, such as the Ras activation of Map3K1 exhibits hypersensitivity, whereas the branch linking Map3K7 to NFκB inhibition is linear and robust to perturbations - perhaps critical information for identifying potential drug targets. The mechanistic insights and predictive power are much higher than what can be obtained from purely data-driven models or simulating purely pathway-derived models.

How does this relate to large multi-scale models? Covert and colleagues [7] were able to develop a whole-cell model of the bacterium *Mycoplasma genitalium* that accounts for all molecular components and their interactions, from electrolytes and metabolites to proteins and ribosome assemblies. This highly complex model was constructed by coordinating sub-models for each of 28 classes of cellular processes, a majority of which were mathematically represented by logic-derived models of some sort (see chapter 3 of supplement S1 in [7] for details). The software and methods developed by Saez-Rodrigues and colleagues [8] make a strong statement about the power of sophisticated logic-derived models for systems such as mammalian cells, where large parts of the molecular networks are not well understood, incomplete and with unknown parameters. But the bottom panel in Figure 2 of [8] provides for both a reality-check of current capabilities and a hint of things to come. Certain behaviors of the studied system (for example, the NFκB oscillations) can be captured only by using the logic-derived differential equations (a method that is considerably more computationally expensive, and which involves many additional arbitrary parameters). This may come as no surprise. Practitioners of detailed, quantitative, validated models have preached for a long time the importance of non-linear dynamics of intracellular molecular interactions, especially in signaling networks, but often also in metabolic or gene regulatory networks (in fact, these classifications of networks are increasingly blurred nowadays). Detailed studies have shown that parts of these networks can act as modules with distinct dynamical features (threshold, hysteresis, oscillatory instability, switch-like instability, and so on) [15]. Such emergent properties may be due not only to network topology, but to the detailed kinetic rate laws and quantitative parameters. To complicate matters further, impedance effects sometimes change individual module behavior when multiple modules are connected to each other.

In fact, much more is swept under the rug than we have alluded to so far. Even the simple cartoon diagram shown in Figure 1a embodies more information than the simple causal links captured by the logic models shown in Figure 1b. Multiple phosphosites can create a combinatorial complexity of regulatory actions, and difficulties in mapping functional states to measured observable quantities. Compartments, scaffolds, and diffusion create spatial inhomogeneities and microdomains, which have critical functionality in many eukaryotic cells (often also in prokaryotes) [16]. And to top it all off, there has been increasing evidence that parts of the cellular machinery employ fleeting, non-stoichiometric, pleiomorphic assemblies of molecules to carry out vital processes [17]. Many novel methods and algorithms have been developed in recent years by the 'bottom-up' modelers and experimentalists to tackle these problems: rule-based [18] and network-free models [19], spatially resolved models with continuous representations (partial differential equations-based) [20] or with discrete representations (particle-based stochastics) [21], as well as refinement of methods long used in mathematical biology, such as agent-based simulation methods and constraint-based models. New theoretical methods and software applications continuously appear in different areas related to modeling, ranging from network-based approaches for predicting missing pathway interactions [22] to multi-level rule-based modeling [23].

Does this detract from our praise of the advances in logic-derived models discussed above? No. To the contrary, this is why we are really excited. Let us return to those logic-derived ordinary differential equations (ODEs and how they (can) relate to the other side of the field. Of course, they are phenomenological constructs of whatever arbitrary mathematical form is being provided (in this case Hill-type equations, which can capture a variety of common non-linear relationships with only two parameters). But such mathematical approximations are sometimes the starting point for discovering the underlying mechanism. In what is arguably one of the most influential modeling works related to biology, almost exactly 100 years ago Leonor Michaelis and Maud Menten used a phenomenological equation to fit the experimental measurements of the initial velocity of the invertase-catalyzed reaction (at time zero, when no product has formed yet, the reaction can be simplified and modeled as being irreversible). Based on that approximation, they posited that the enzyme activity could be explained by mass-action kinetics involving an intermediary reaction complex - the fundamental mechanism of enzymatic catalysis that was confirmed three decades later [24]. The fact that explaining certain qualitative characteristics requires the ODE-based formalism is the perfect starting point for directing new detailed investigations of potential mechanistic hypotheses.

Moreover, if we can modify a logic-derived model and end up with a differential equations-based model, why not jump over the fence and use what is available in the world of kinetic models? For starters, much more powerful optimization algorithms and tools have been developed in that domain [25]. Taking advantage of these would be trivial if models could be exported into a community standard format such as Systems Biology Markup Language (SBML) [26]. And if support for this exchange format were implemented in reverse, too, one could import pre-existing detailed kinetic models into software that deals with logic formalism as just another form of prior knowledge for those interactions where such information already exists (for reference, as of this writing, the Biomodels database makes available 154,456 kinetic relationships between 133,559 molecular species in the curated branch).

It sounds trite to say that we need to use multiple approaches and tools in order to build truly complete and accurate cellular models. We are getting closer not only to integrating multiple logic-based formalisms easily, but also to crossing over into kinetic, spatial, rule-based models, and more. And the experimental data required for building all these different types of computational models at different scales and levels of detail will have to come from both 'small science' and 'big science' [27].