Probabilistic Inference with Polymerizing Biochemical Circuits

The magnitude of this effect will depend on the rate at which A and B are incorporated into polymers (k in Figure 2A) and the number of polymers being assembled, which is bounded by the abundance of T monomers. The rate of polymer assembly will in turn influence the rate of production of memory molecules. If A and B are abundant and T is small, as in Figure 3 and Figure 4 where there are 20 Ts, then a single switch from, say, A to B can produce at most 20 A-To-B memory molecules. The concentration of T, therefore, influences the magnitude of the circuit’s memory (measured by copies of memory molecules), and this in turn affects the sensitivity and noise in the binding of integrators (in all of the above simulations, integrators are assumed to be fixed at 400 copies). The parameters governing polymer assembly, therefore, shape how responsive the circuit can be to environmental change.

The abundance of the circuit’s components relative to those of A and B also matters. In all of our simulations, A and B were far more abundant (5000 copies) than the number of polymers being assembled (10 Ts), and we chose an appropriately small polymerization rate constant (k=0.0001; recall that k is the rate constant governing the addition of monomers to the end of a polymer). If A and B were in low abundance and/or the polymerization rate was really slow, then environmental switches would go undetected. On the other hand, if the polymerization rate was too high and/or the concentration of T too high, the circuit could in theory consume all the “nutrients” (using these to construct very long polymers), and therefore become insensitive to different durations (i.e., be unable to produce duration-encoding memory molecules). Naturally, the stability of the memory molecules produced through polymerization will determine how far back the circuit will “remember” environmental switches.

While the periodic and “sticky” environments we have explored produce very different sequences of switches, both fall within our definition of Markov environments. Moreover, we have assumed that each environment follows the same set of transition probabilities throughout. We next asked how the circuit would behave in environment’s where there is a sudden change in the way that the environment changes (similar to the “meta-changing” environments described in [13]).

In the environment shown in Figure 6, for example, the initial sequence of switches (time 0–500) is drawn from a periodic environment, πAB=πBA=0.95, and the remaining sequence from a “sticky” B environment, πAB=0.85,πBA=0.05 (Figure 6, time 500 through end). The circuit adapts to the new set of statistics, as seen in the changing curve of fraction of active integrators (Figure 6C, starting at time 500). The resulting polymers also correctly reflect the environment’s two distinct segments (Figure 6E). Naturally, the rate at which the circuit will adapt to such changes in the environment’s underlying dynamics will depend on how quickly the memory molecules are degraded.

The probabilistic information obtained from the history of switches may be used to regulate other processes within the cell, such as a response to a specific environment. For this to be possible, as some cognitive scientists have argued, the information has to be “accessible” [42]. For accessibility, it is not sufficient to only have the system “contain” the information in its states in an information-theoretic sense (as was shown for memory in B. subtilis [16]). Rather, the information has to be instantiated in a form that is usable by other molecular processes within the cell.

We next show how the probabilistic information about the environment’s history can be used to regulate an environmental state-specific program, as shown in Figure 7A. In this scenario, the organism is probabilistically presented with a toxin, and (1) an anti-toxin program is required in order to increase chances of survival or growth, (2) the anti-toxin program takes time and substantial energy to produce (so it cannot be kept on at all times). Depending on the frequency of the toxin, the timescale of environmental change, and the associated metabolic costs, there are scenarios where it makes sense for organisms to anticipate the onset of the toxin and produce the program in advance ([13]; see also [43] for a seminal discussion of such trade-offs in changing environments).

We consider an abstract version of this problem in which a protein associated with the B state, B-Program (akin to the anti-toxin program), has to be produced prior to the onset of B in a periodic environment. We assume that the ideal expression profile of B-Program would one where B-Program is produced in advance, reaches its peak at the onset of B, and then degraded (as shown in Figure 7A). Moreover, as the circuit experiences more periodic switches, we expect the regulation of B-Program to become more pronounced, reflecting the fact that more “data” about the environment’s transition probabilities has been obtained.

To create this anticipatory behavior, we can utilize the posterior probability of encountering B, encoded in the integrator-memory molecule complexes. We designed a circuit that regulates B-Program in proportion to its posterior probability as encoded by these complexes (Figure 7B). This posterior probability is encoded in the fraction of A-To-B integrators that are active, i.e., bound by the A-To-B memory molecule. When A is present, it can bind these active integrators, forming a complex that produces B-Program (Figure 7B). However, when A binds an A-To-B integrator that is repressed (i.e., bound by A-To-A memory molecule), the resulting complex will degrade B-Program. When B is present, a symmetric circuit that uses the B-To-A integrator to enact the same logic with B-To-A and B-To-B memory molecules.

Crucially, the rate constants governing this circuit must match the timescales of change in the environment. If the desired behavior is to have the A-specific program (for example) reach its maximal expression prior to the onset of A, the circuit must be tuned to the average duration of A and B states. This “tuning” is reflected in the choice of rate constants. The rate constants for these interactions—which determine the binding of A and B to integrators, and the production/degradation of B-Program—have to be assigned in a way that produces the desired “sawtooth” expression profile shown in Figure 7A. We used an optimization procedure to set these rates (see Appendix A for details). With the resulting rate constants, the circuit indeed regulates B-Program in sawtooth-like fashion, where the regulation becomes more pronounced with increasing exposure to switches (Figure 7C). This behavior is consistent with the idea that more experiences of the environment lead to more confident estimates of transition probabilities (by accumulation of memory molecules), and that more confident estimates produce sharper responses to switches. Note that due to the regulation of B-Program, some of the A to B molecules remain bound to integrators during a switch, and are thus not removed, resulting in a mixture of A and B molecules in the next state. The number of molecules that can be carried over in this way is bounded by the number of integrator molecules.

We have described a polymerizing circuit that performs probabilistic inference in a Markov model. Our analysis of the requirements of real-time inference in the Markov model led us to look for ways to biochemically realize two somewhat counter-intuitive operations: directional counting and normalization-sampling. The first was performed through polymer assembly, relying on the polarity of polymers and their constituents. The second was performed using a scheme where a pair of proteins bind to a third “integrator” protein in mutually exclusive fashion. With these two operations, the circuit can be used to regulate another biochemical process of interest, such as an environmental state-specific program.

The resulting circuit has several interesting properties. First, this circuit exhibits predictive behavior in a dynamic environment without relying on mutation and natural selection across generations. While mutation and selection can, from a mathematical perspective, implement solutions to probabilistic inference problems [44], and Darwinian evolution and inference algorithms have been argued to be conceptually similar strategies for solving the same basic problem [45], mutation and selection are nonetheless biologically distinct processes that operate on longer timescales than ontogenetic ones. The former operate on longer timescales; the latter can occur, theoretically, during the “lifetime” of a single cell experiencing environmental change. The ontogenetic behavior we explored is therefore distinct from laboratory evolution experiments with microbes or in silico-evolved circuits that were selected to fit a particular environment (although several previous studies have presented such experimental results as evidence of “predictive” cellular behavior [46,47], as argued in [30]).

Second, our circuit consumes the “signal” (A and B, in our case) that it anticipates. These molecules become part of the circuit (organism) through the process of polymer assembly. While we have referred to “the environment” as if separate from the circuit, this circuit actually alters the concentration of A and B (the magnitude of the effect depends on the relative abundances of nutrients and the rate of polymer assembly). This demonstrates the blurry line between metabolism and signal transduction [48].

Since the circuit records the history of change in polymers, this opens up possibilities for molecular mechanisms that utilize these sequences by interacting with the polymers. A variety of molecular mechanisms, such as kinesin motors that walk along microtubules—which have been intensively studied experimentally [49] and theoretically [31]—can be a source of inspiration. Microtubules can also be regulated through myriad post-translational modifications to tubulin proteins, which have been viewed as a kind of combinatorial “tubulin code” [50,51,52]. These modifications can affect the localization and size of microtubules by tuning the binding of microtubule-regulating proteins. Such mechanisms can also be explored as forms of biological information-processing.

A theoretical study of these mechanisms would benefit from further development of formal languages for representing and simulating combinatorial biochemistry, such as Kappa [40]. Kappa, for example, currently does not explicitly model physical space, which raises a challenge: how to capture more of the ways in which physical space constrains and shapes polymerization—in a dividing cell, for instance—while keeping the formal language sufficiently abstract so as to be interpretable, and the simulations tractable? Computational tools for visualizing polymer assembly, through time and space, could also help the study of polymerization as a form of information-processing.

Several open questions about polymerization and inference also remain. It would be interesting to explore how the circuit design we have proposed can be adapted to more complex environments. A recent study has investigated how inference in hidden Markov Models can be performed using CRNs [53], and it would be worth exploring whether a polymerization scheme for real-time inference of the sort we have developed could be used for these more complex Markov models that have latent states. Similarly, it may be fruitful to explore how circuits could cope with “meta-changing” environments whose dynamics change according to yet another unobserved stochastic process [13]. Another avenue would be to explore the energetic costs of such inference-performing circuits, and how they might be integrated into a larger cellular model that accounts for cell division, growth, and the cell’s self-producing (“autopoietic”) capacity [54,55].