3.1. Explanandum Context

Escherichia coli (E. coli) is popular model organism in biological research. It is very sensitive to small chemical changes over a very large range of background concentrations. It also has a simple and well-understood signal transduction network (Wadhams and Armitage Reference Wadhams and Armitage2004).

E. coli manages two kinds of motion (Berg Reference Berg2004). It runs by rotating its flagellar motor counterclockwise. This aligns all of its flagella into a synchronized bundle, resulting in movement in a straight line for about 1 second. E. coli also tumbles by rotating its flagellar motor clockwise. This breaks flagellar alignment, and the asynchronized flagella produce stationary changes of direction lasting for about 0.1 seconds. E. coli randomly reorient after each tumble. Moreover, while these tumbles occur with regular frequency, E. coli with higher concentrations of CheR protein tumble more frequently (Spudich and Koshland Reference Spudich and Koshland1976).

E. coli’s motion in a uniform external environment resembles a random walk. E. coli has no ability to control or select its direction of motion, and its straight runs are subject to Brownian motion because of eddies. However, in the presence of a chemical attractant—amino acids such as serine or sugars such as maltose—E. coli taxis toward the attractant. This taxi behavior involves less frequent tumbles, leading to longer runs and so gradual motion toward the attractant. (There is an opposite behavior for repellents such as metal ions and leucine.)

The biomolecular mechanism for E. coli chemotaxis is well understood. When an environmental attractant (ligand) attaches to a receptor, the receptor lowers the activity of the CheW-CheA protein complex. Less activity from this complex reduces the rate of CheY phosphorylation, which results in less phosphorylated CheY diffusing to the flagella. Because CheY induces clockwise rotation of the flagellar motor, the outcome is less frequent tumbling.

3.2. Explanandum Question

Alon and colleagues have experimental verification that, in the presence of a chemical attractant mixed uniformly into the environment at a constant concentration, E. coli chemotaxis is perfectly adaptive: after a brief period of decreased tumbling frequency, the frequency of E. coli tumbling increases toward and returns to the exact frequency before the introduction of the attractant (Alon et al. Reference Alon, Surette, Barkai and Leibler2009). The effect of the attractant, accordingly, becomes entirely forgotten despite its continuing presence.

The biomolecular mechanism for the adaptiveness of chemotaxis for E. coli is also well understood. Sometime after a new attractant has been detected by receptors, the lower activity of the CheW-CheA complex induces less CheB activity. This reduces the rate for removing methyl groups from the CheW-CheA complex. Because there is continual methylation of the CheR receptor, CheW-CheA methylation increases. More methylation means more CheW-CheA activity, which in turn induces more CheY phosphorylation. This eventually results in more phosphorylated CheY diffusing to the flagellar motor, which increases clockwise motor rotation and thereby increases the tumbling frequency. Figure 2 visualizes this mechanism (along with some omitted details).Footnote ¹

Figure 2.

Mechanism sketch for chemotaxic adaptation in E. coli. Arrows represent “activation,” whereby more activity at the tail produces more activity at the tip. “Flat-tipped” lines represent “inhibition,” whereby more activity at the tail produces less activity at the tip.

Alon and colleagues (Reference Alon, Surette, Barkai and Leibler2009) have further experimental verification that this perfectly adaptive chemotaxis of E. coli is robust across ranges of CheR concentrations 0.5 to 50 times higher than concentration levels in “wild type” E. coli.Footnote ² This is the explanandum of interest: why is the perfect adaptation of E. coli chemotaxis, in the presence of a well-distributed chemical attractant, robust to CheR protein concentrations?

4.1. Dynamical Modeling

I call the first strategy dynamical modeling. This strategy begins by constructing a chemotaxis network for E. coli. This network represents more than the mechanism for E. coli chemotaxis, by also including specific biochemical details about when and how relevant proteins affect each other. For example, Barkai and Leibler (Reference Barkai and Leibler1997) construct a model according to which, among many other specifics, CheB demethylates only the active form of the CheW-CheA complex, and CheR works only at saturation. Here is part of the network, simplified through omission of kinetic constants associated with each reaction and followed by translations into English:

(1)$\begin{array}{}& {\overline{E}}_m^{*}{E}_m^{*}+B\leftrightarrows \left\{E_m^{*}B\right\}\to E_{m-1}^{*}\mathrm{.}\end{array}$

(2)$\begin{array}{}& {\overline{E}}_m^{*}{E}_m^{*}+R\leftrightarrows \left\{E_m^{*}R\right\}\to E_{m+1}^{*}\mathrm{.}\end{array}$

(3)$\begin{array}{}& E_m^u+L\leftrightarrows E_m^o\mathrm{.}\end{array}$

The dynamical modeling strategy proceeds by constructing a dynamical model—typically a set of differential equations—from the reaction network (see Jones and Wolkenhauer Reference Jones and Wolkenhauer2012). One then demonstrates, via mathematical proof or simulation, that this model predicts perfect adaptation in the presence of a well-distributed chemoattractant for CheR concentration values varying over several orders of magnitude. The demonstration supports the inference that E. coli chemotaxis exhibits robust perfect adaptation because of its biochemical specifics.

Bechtel and Abrahamsen (Reference Bechtel and Abrahamsen2010) call the product of this strategy a dynamical mechanistic explanation. I set aside the issues of whether the dynamical modeling strategy produces explanations and of whether the dynamical models are mechanistic. I also do not attempt to identify the various ways in which Barkai and Leibler’s dynamical model—and its associated reaction network—is abstract. Instead, I focus on the ways in which models associated with other explanatory strategies are more abstract and, in particular, on the abstractions present in these other models that are absent from Barkai and Leibler’s model.

4.2. Topological Analysis

I call the second explanatory strategy topological analysis. This strategy begins by identifying all possible network motifs capable of predicting robust perfect adaptation. These motifs, like the reaction networks for dynamical modeling, are meant to explain the robust perfect adaptation of E. coli chemotaxis. Yet, unlike the reaction networks, these motifs are minimal: they contain the fewest possible nodes and links that suffice for robustly perfectly adaptive chemotaxis. The procedure for identifying all possible motifs of this sort is brute computational search. It turns out that there are exactly three, each of which has exactly three nodes and no more than three links (Ma et al. Reference Ma, Trusina, El-Samad, Lim and Tang2009).

The topological analysis strategy proceeds by identifying a reaction network known to predict robust perfect adaptation for E. coli chemotaxis. This strategy thereby relies on the dynamical modeling strategy, but only for mathematical results. The biochemical details of the chosen network turn out to be largely irrelevant because the topological analysis strategy proceeds by demonstrating that a reduced form of the chosen network is topologically equivalent to one of the network motifs. These reduced forms group mechanistic details together into functionally equivalent modules and consider interactions among modules rather than among entities. They are thereby less sensitive than reaction networks to variations of detail.

Consider, for example, one of the motifs Ma and colleagues (Reference Ma, Trusina, El-Samad, Lim and Tang2009) discover for E. coli chemotaxis (see fig. 3a). The motif represents some unspecified input (dashed arrow) activating an “input-receiving” node A; a negative feedback loop in which A inhibits a “buffering control” node B that, in turn, activates A; and an “output-transmitting” node C being activated by A and, in turn, producing some unspecified output.

Figure 3.

a, Motif for E. coli chemotaxis. b, Subnetwork relative to which the module controlling motor rotation is an aggregation.

Ma and colleagues demonstrate that Barkai and Leibler’s (Reference Barkai and Leibler1997) reaction network for E. coli chemotaxis reduces to, and is thereby topologically equivalent to, this network motif. Barkai and Leibler’s reaction network involves a ligand activating the methylated CheW-CheA complex, continual methylation of this complex by CheR along with a negative feedback loop in which the complex inhibits its own demethylation by inhibiting CheB phosphorylation, and the methylated complex affecting tumbling frequency by affecting CheY phosphorylation.

The topological analysis strategy infers, from the topological equivalence between a network motif and a reaction network known to predict robust perfect adaptation for chemotaxis, that E. coli chemotaxis exhibits robust perfect adaptation because of the topology for its chemotaxis network. Huneman (Reference Huneman2010) calls the product of this strategy a topological explanation. Regardless of whether such analyses are explanatory, they are topological by virtue of demonstrating some consequence about the topological properties of a network. So even if the mechanistic details of E. coli’s chemotaxis network were different, and even if the biochemical specifics of the original reaction network were different, the product of the topological analysis strategy would remain the same, provided that the alternative networks preserve topological equivalence with the originals (see Jones Reference Jones2014).

The topological model driving this second explanatory strategy is more abstract than the dynamical model driving Barkai and Leibler’s explanatory strategy. The topological model contains two aggregations: from a subnetwork of interactions among unmethylated CheW-CheA, CheR, and CheB to a module that controls methylation and from another subnetwork of interactions among CheY and CheZ to a module that controls motor rotation (see fig. 3b). I interpret these abstractions as aggregations because they are many to one: they group together entities and activities working together for some common function.

The model also constructs two structurations from these aggregations: from the methylation-controlling module to buffering control node B and from the motor-rotation module to output-transmitting node C. I interpret these as structurations rather than generalizations. The modules are well defined without reference to their broader role. But Ma and colleagues speak of the nodes relationally, providing functional characterizations for each: C is the “output-transmitting” node, while B is a “dedicated regulation node … that functions as a ‘buffer’ [for A]” (Reference Ma, Trusina, El-Samad, Lim and Tang2009, 764). So conceptualized, the abstraction from module to node increases relationship arity, from module to module for an output or for an input receiver. For similar reasons, I interpret three further abstractions as structurations: from ligand to input (for a receiver node), from tumbling frequency to output (of a transmitting node), and from methylated CheW-CheA complex to input-receiving node A.

4.3. Organizational Design

I call the third explanatory strategy organizational design. This strategy begins with a formal proof to the effect that systems exhibit robust perfect adaptation if and only if they satisfy the characteristic equation for integral feedback control (IFC). The proof is purely mathematical, well known from control engineering theory in contexts involving mechanical systems that exhibit IFC such as thermostats. There is debate about whether the proof holds only for deterministic systems (Briat, Gupta, and Khammash Reference Briat, Gupta and Khammash2016). I set this aside, focusing on the explanatory strategy rather than its effectiveness.

The organizational design strategy infers, from the equivalence between robust perfect adaptation and IFC, that E. coli chemotaxis exhibits robust perfect adaptation if and only if it satisfies the characteristic equation for IFC. It also adds that E. coli chemotaxis exhibits robust perfect adaptation because it satisfies the characteristic equation for IFC: the organizing principle, IFC, is supposed to explain the phenomenon of interest.

This strategy invokes neither mechanistic specifics about the chemotaxis network for E. coli nor topological details about the structure of that network. The strategy takes the explanandum phenomenon as given, uses a mathematical equivalence result to identify a principle both necessary and sufficient for the phenomenon, and infers that the phenomenon obtains by virtue of the principle. The strategy thereby has affinities with explanatory strategies that appeal to organizing principles (Green and Wolkenhauer Reference Green and Wolkenhauer2013).

The workings of this strategy are difficult to follow without understanding the sense in which E. coli exhibits IFC. So consider figure 4, a standard block diagram for visualizing IFC. An input acts on a system (near the top right), generating an output and a “measure signal.” A comparison between this signal and a control signal generates an “error signal.” A controller (square with vertical stripes) then transforms the error signal into a perturbation signal designed to modulate the system’s activity toward some desired output state.

Figure 4.

Block diagram for integral feedback control.

E. coli exhibits IFC organization as follows. The system is the methylated CheW-CheA complex. The input is the ligand, which interacts with this complex; the output is the activity of the complex on CheY and everything downstream of that activity. The “measure signal” is the activity of the complex on CheB, which determines the rate of CheB phosphorylation. The “control signal” is the maximum rate at which CheR methylates the CheW-CheA complex, determined by the total concentration of CheR. There is a comparison between this rate and the rate of CheB phosphorylation. Nothing performs this comparison. But there is an “ideal” balance among the rates determined by the tumbling frequency before ligand exposure, and the “error signal” is the difference between this “ideal” balance and the actual balance. The controller is phosphorylated CheB, and the perturbation signal is its activity in demethylating the CheW-CheA complex.

Yi and colleagues’ organizational design strategy for explaining the robust perfect adaptation of E. coli involves a model that is abstract in all the ways its cousin topological model is abstract and more. Some abstractions are the same: the motor rotation module aggregation and the structuration output from this module,the structuration input from the ligand, and the structuration system from the methylated CheW-CheA complex. Others differ. There is an aggregation of the module controlling phosphorylation and dephosphorylation of CheB and the structuration controller from this module. There are several generalizations as well: a control signal, abstracted from total concentration of CheR; a measure signal, from the activity of methylated CheW-CheA on phosphorylated CheB; and a perturbation signal, from the activity of phosphorylated CheB on methylated CheW-CheA. Hence, disregarding judgments about abstraction types, Yi and colleagues’ model targets its abstractions to mechanistic details different from those of Ma and colleagues’ topological model.