Memory in microbes: quantifying history-dependent behavior in a bacterium

Memory is usually associated with higher organisms rather than bacteria. However, evidence is mounting that many regulatory networks within bacteria are capable of complex dynamics and multi-stable behaviors that have been linked to memory in other systems. Moreover, it is recognized that bacteria that have experienced different environmental histories may respond differently to current conditions. These “memory” effects may be more than incidental to the regulatory mechanisms controlling acclimation or to the status of the metabolic stores. Rather, they may be regulated by the cell and confer fitness to the organism in the evolutionary game it participates in. Here, we propose that history-dependent behavior is a potentially important manifestation of memory, worth classifying and quantifying. To this end, we develop an information-theory based conceptual framework for measuring both the persistence of memory in microbes and the amount of information about the past encoded in history-dependent dynamics. This method produces a phenomenological measure of cellular memory without regard to the specific cellular mechanisms encoding it. We then apply this framework to a strain of Bacillus subtilis engineered to report on commitment to sporulation and degradative enzyme (AprE) synthesis and estimate the capacity of these systems and growth dynamics to ‘remember’ 10 distinct cell histories prior to application of a common stressor. The analysis suggests that B. subtilis remembers, both in short and long term, aspects of its cell history, and that this memory is distributed differently among the observables. While this study does not examine the mechanistic bases for memory, it presents a framework for quantifying memory in cellular behaviors and is thus a starting point for studying new questions about cellular regulation and evolutionary strategy.

Your average bacterium is unlikely to recite π to 15 places or compose a symphony. Yet evidence is mounting that these ‘simple’ cells contain complex control circuitry capable of generating multi-stable behaviors and other complex dynamics that have been conceptually linked to memory in other systems. And though few would call this phenomenon memory in the ‘human’ sense, it has long been known that bacterial cells that have experienced different environmental histories may respond differently to current conditions [1]–[3]. Though some of these history-dependent behavioral differences may be physically necessary consequences of the prior history, and thus some might argue insignificant, other behavioral differences may be controllable and therefore selectable and even fitness enhancing manifestations of memory.

In this paper we take the potentially controversial view that history-dependent behavior, whether short or long term, controlled or incidental, reflects a form of memory [4]–[6]. Because bacterial dynamics at every level of resolution operate within the limitations and potentials of nonlinear physical and biochemical dynamical systems, they must exhibit at least very short-term transient memory, and potentially longer term memory. The type of memory (and its significance) depends on which features of cell history are ‘remembered’, and at what resolution; whether or not the system eventually ‘forgets’ its past, and if so, how long this forgetting takes; the mechanisms in the cell responsible for memory storage, encoding, and retrieval; and whether or not this memory provides a fitness advantage in a natural environment. In cellular systems, environmental memory has been noted to be inherent in everything from the selective history of mutation, epigenetic inheritance via chromatin modification in neurons and DNA methylation in chemotaxing bacteria [7], genetic and epigenetic phase variation mechanisms controlling surface features of pathogenic bacteria [8], [9], cellular proliferation and survival in the immune system, and in switch-like feedback systems in regulatory networks spanning signal transduction, metabolism and gene expression [10]–[21]. There is also a growing body of work focusing on synthetically designing and constructing network motifs and systems that are capable of showing some types of dynamic memory [22], [23].

These and many other studies in synthetic and natural systems suggest that even the simplest first-order chemical reactions have at least transient memory of initial conditions, and more complex mechanisms involving history-dependent changes in the concentrations, states and localization of proteins and other regulatory network elements can encode a wide range of input information and store it for amounts of time ranging from minutes to days or longer [4], [16], [24], [25]. The state dynamics of such systems contain the memory of past controlling inputs, and even of past environmental conditions if one is to interpret more broadly [5], [26].

In metazoans, the ability of somatic cells to remember their fates is key to development and thus to organismal fitness. The same can be said for other types of metazoan cells like those found in the immune system that use a memory of past states to modify future behavior. In principle at least, memory, whether short- or long-term, can feasibly confer an evolutionary advantage in microbes as well. For instance, Hoffer et. al. suggest that in E. coli a form of ‘memory’ of past phosphate limitation leads to a faster response to successive periods of phosphate limitation, and that this faster response may be survival enhancing [5]. It has also been suggested that pathogenic bacteria use cross-talk encoded memory to balance the demands of immune avoidance with a sequential, compartment to compartment infection lifecycle [8], [9]. More abstractly, the dynamic implementation of cellular behaviors can be viewed as a selected, ‘winning’ (or at least stable) strategy in an evolutionary game [12], [27]. In game theory, information creates advantage [28]–[30], and information about the past as well as the present creates even greater advantage. Thus if bacterial cells are able to store information about past experience in some type of memory, and use this memory to modulate their behavior, this opens up the possibility of playing game strategies with memory, a provably superior family of strategies compared to those without memory [31]–[35]. Even if the memory capacity of the system is short term, but on the order of environmental fluctuations, it could conceivably impact fitness and therefore play a role in an evolved adaptive behaviour [28].

Given the potential ubiquity and significance of bacterial memory, we propose that quantifying history dependent behavior in microbes could be an important piece of the puzzle of bacterial regulation, survival strategy, and evolution. To this end, we developed an information-theory based conceptual framework for thinking about and measuring both the persistence of memory in microbes and the amount of information about the past encoded in these dynamics. This method produces a phenomenological measure of cellular memory without regard to the specific cellular mechanisms encoding it. We then applied this framework to the bacterium B. subtilis. B. subtilis presents an excellent model organism for this study because of its exquisite sensitivity to environmental conditions, its known mechanisms of bistability and other hysteretic switch-like regulatory stress response mechanisms and architectures, and its developmental decision to sporulate that strongly resembles eukaryotic memory-associated processes determining developmental cell fate ([10], [36]–[40], Fig. 1). Also, certain aspects of B. subtilis behavior, such as spore coat composition, have already been associated with environmental memory [41]–[43], and though much suggests that there should be memory, how these response dynamics depend on past conditions prior to application of a stress has not been systematically examined.

In our experiments, we quantified the ability of three B. subtilis stress response systems–sporulation, degradative enzyme synthesis, and growth-to ‘remember’ 10 distinct cell histories prior to application of a common stressor. We chose to observe commitment to sporulation (via reporter fusion to PSPOIIE) because the sporulation decision is bistable, and bistability is associated with memory [9], [11], [16], [44]. We added the reporter for degradative enzyme synthesis (measured by a fluorescent reporter fused to the AprE promoter) because though it shares many common controllers with sporulation, its expression pattern is quite different and not believed to be bistable or probabilistic. We wondered whether any history-dependence in sporulation control would be mirrored in AprE control. Finally, we chose to observe growth (as measured by OD600) because it is perhaps the most accessible measure of cellular health and fitness and is an integrator of many other aspects of cell function, thus it may show interesting differences depending on cell history. One can imagine that there might be a strong fitness incentive toward memory in B. subtilis. If cells could use a memory of past conditions to ‘predict’ future conditions, and delay sporulation, an expensive process, if the environment is likely to improve or accelerate sporulation if the starvation period is likely to be long, they might improve their odds for long-term survival.

A complete quantification of biologically relevant memory would involve first perturbing the cell with all possible sequences of complex environmental inputs it might experience in the wild in each of its growth modes, then measuring all cellular responses to these perturbations, and, finally, quantifying the degree and distribution of history-dependence in these responses.

Here we assume a simple approximation of this scenario, in which each sample of a biological system is subjected to one of many conditions prior to time t0, and then observed in a common condition after t0 (see Fig. 2 and Definition (1) in Appendix S1 in Supplementary Information). We call this an ‘adaptive’ memory experiment because it roughly simulates a temporal shift in the environment requiring adaptation or acclimation, and to differentiate it from the more classical memory experiments in physics, engineering and cell biology designed to identify hysteretic loops [45]–[47]. While we do not identify such loops here, multistability is suggested by the appearance of long term memory in our experiments. More complex environmental history trajectories could feasibly unravel more memory effects.

We are interested in whether past conditions can be inferred from observations of behavior in current conditions. The assumption here is that history-dependent behavior is a manifestation of memory, and that the better the possible inference about prior conditions from current measurements, the more memory there is within the system.

To quantify this intuitive concept of history-dependence as memory, we use concepts from information theory [48] in the tradition of Landauer's use of informational entropy to estimate human memory capacity [49], and the extensive body of work characterizing memory in individual neurons [50]–[53].

By interpreting the random variable Y as behavior in current conditions, and the random variable M as past cellular history prior to time t0, the mutual information I(M;Y) = H(M)−H(M|Y) of M relative to Y provides a measure of memory in informational entropy bits (see [48], Fig. 3, and Definition (2) in Appendix S1 for details, including the definition of informational entropy H). Roughly speaking, from this perspective I(M;Y) captures how much uncertainty about past conditions can be reduced by observations of behavior in current conditions. Worded differently, I(M;Y) captures how much information about past conditions can be inferred from observations of behavior in current conditions. The better the possible inference about prior conditions (and thus the higher the bit count of I(M;Y)), the more memory there is within the system.

Memory, or history-dependent behavior, can manifest across multiple time scales. Short term, or transient, memory is stored by the system for some time, and then ‘forgotten’ (see Fig. 4a,d). Systems may also have either ‘effective’ long term memory if the transient dynamics are long compared to environmental fluctuations, or ‘true’ asymptotic memory if the stationary state of the system depends on initial conditions, as occurs in nonlinear systems with multiple attractors (see Figs. 4b,c,and e). For an example of the latter, the state of a bistable switch encodes an asymptotic memory of the last switching event.

Because in many systems the significance, mechanistic origin, and function of memory likely depends on how long it lasts, and in particular whether it can be classified as short-term or long-term, we distinguish between the two types of memory and quantify them separately. From an information perspective, we say that an external observer of an adaptive memory experiment with a priori knowledge of the probability distribution over cell histories detects short-term memory in this system if observing measurements of some fraction of the short-term behaviour of the system after time t0 leads to a reduction in uncertainty about the history of the system prior to time t0. In this case, we say that the cells exhibit Itrans(M;Y; ttrans)≡I(M;Y(t = t0:t0+ttrans)) bits of short term memory in the observable Y over the period from t0 to t0+trans, where ttrans is a time before the signal approaches its steady state (Definition (4) in Appendix S1). Likewise, long-term memory is detected if observing measurements of the system behavior near an apparent steady state after time t0 leads to a reduction in uncertainty about the history of the system. Here we say the cells exhibit Iasym(M;Y)≡I(M;Y(t = t0+tasym:∞)) bits of long term memory in the observable response Y during the experiment, where tasym is the time it takes for the signal to settle (Definition (3) in Appendix S1).

The above metrics for short term and long term memory are absolute measures, in that they give a bit count for an answer. Though these absolute numbers can be useful, it is also useful to measure memory in relative terms, compared to the total amount of memory that could be observed in a perfectly retentive system given the limitations of the experiments. To address this issue, we define short-term memory fidelity to be Ptrans(M; Y; ttrans)≡I(M;Y(t = t0:t0+ttrans))/H(M) and long-term memory fidelity to be Pasym(M;Y)≡I(M;Y(t = t0+tasym :∞))/H(M), where H(M) is the entropy over all the past conditions that were applied in the experiment. These normalized mutual information metrics, measures between 0 and 1 of the fraction of uncertainty about the past conditions tested that is reduced by knowledge of future cellular response, have also been called the coefficients of constraint [54] (see Definition (5) in Appendix S1).

In addition to analyzing each observable individually, we are interested in calculating the short and long term memory exhibited by the combined behavior of multiple observables. To do so, the above definitions are easily extended to the case of multiple observables by letting Y be a vector Ȳ = (Y 1,…,Yn) and calculating Iasym(M;(Y1,…,Yn)) and Itrans(M;(Y1,…,Yn); ttrans) and the memory fidelity of each. This combined-memory estimation is interesting because it allows one to address the question of whether combining information from multiple read-outs leads to extra memory beyond what is present in any of the individual read-outs, and if so, how much. This issue is related to the size of the memory, and the dimension it occupies within a cell's state space.

An inequality governing the informational entropy of a vector pair of variables (X,Y) is as follows: max(H(X),H(Y))≤H(X,Y)≤H(X)+H(Y) [54]. Thus, we know that the memory exhibited by any pair of observables must be greater or equal to the bit count of the most retentive pathway of the pair, and less than or equal to the sum of the bit counts of the two pathways. If two pathways are controlled independently, their combined behavior could produce the upper limit on memory in the higher-dimensional space, whereas if the pathways are controlled by a common signal or if one pathway hierarchically controls the other, the lower limit might be realized. To quantify this concept, we define memory orthogonality between two pathway readouts Y1 and Y2 to be: Memorth(M;(Y1,Y2)) ≡ (I(M;(Y1,Y2))-max(I(M;Y1),I(M;Y2)))/min(I(M;Y1),I(M;Y2)), where M is cell history and I is mutual information. Memorth equals 1 if the two variables combined as a vector yield the upper bound of memory, and 0 if the two variables in combination yield the lower bound (see Definition (6) in Appendix S1).

For the calculations above, listed more formally in Appendix S1 in Supplemental Information, we need to estimate probability distributions over the past cell histories being tested and the responses of the cells to each history. For past conditions/histories, we enforce a uniform probability of observation of each condition by running each experiment (condition i = > response i) a fixed number of times. For responses, we cluster trajectories from the different conditions and the probability of a response is simply the histogram of trajectories over clusters. The probability of prior environment given cluster membership is enumerated in a similar way. Details of the entire analysis algorithm can be found in Materials and Methods.

The above information-based metrics and simple associated analysis algorithm (see Materials and Methods) are useful in that they transform the ‘lay’ questions–“Do cells ‘remember’ past experiences and use these memories to modify future stress response dynamics?” and “If so, is this ‘memory’ short term or long term, and how much is there?”–into well-defined queries about information and uncertainty yielding quantitative estimates of microbial memory in informational entropy bits.

However, any attempt to quantify or qualify memory is fundamentally limited by the possibility of unobservable states (see Fig. 4c), uncontrolled and unobservable inputs, poor choice of input combinations and sequences, and measurement errors and distortions. Here we assume most such limitations, discussed in more detail in Supplementary Information (Section S1), are inherent in the estimation of memory processes and most likely to result in information loss and thus underestimates of the ability of the system to ‘remember’ the cell histories tested by the experimental compendium. Therefore we interpret quantifications of memory within our B. subtilis compendium as lower bound estimates.

Memory experiment on B. subtilis: To test for history dependent behavior–‘memory’-in B. subtilis, we engineered a fluorescently labeled strain of Bacillus subtilis to report on commitment to sporulation and degradative enzyme synthesis: the KEE strain (PspoIIE-gfp, PaprE-dsred cmp, see Materials and Methods for details on strain construction). The spoIIE promoter (PspoIIE), our sporulation reporter, controls expression of spoIIE, a gene encoding a serine phosphatase specifically expressed upon commitment to sporulation and therefore considered a good sporulation commitment signal [55], [56]. The aprE promoter (PaprE), our degradative enzyme synthesis reporter, controls expression of the extracellular protease subtilisin naturally produced by B. subtilis cells at the end of exponential growth [57].

With the KEE reporter strain, we used our framework to estimate, in informational entropy bits, the capacity of these stress response pathways and of the cell growth dynamics to ‘remember’ 10 distinct cell histories prior to application of a common stressor. Specifically, we first grew three replicate cultures in one of two media, Luria Broth medium (LB) or growth medium (GM) [58], to one of five different densities (all still in exponential growth, ranging from OD600 = [0.1:1], see Table 1, where OD600 is the optical density of the culture at 600nM), for a total of ten cell histories. Thus in the first stage of the experiment, a clonal population of cells was divided into 30 groups, each of which experienced one of the 10 cell histories consisting of growth in one of two media to one of five cell densities over a fixed period of time (see Materials and Methods for details).

We chose to combine different media with growth to different densities as our set of cell histories because growth media can impact cell state, as can growth of cultures to different densities over a fixed period of time. Cells deplete nutrients and respond to the environment and its dynamics with changes in metabolic fluxes, post-translational modifications, gene expression, quorum signaling and synthesis of storage compounds. GM medium (also called CH medium) is a rich medium with casein hydrolysate as the sole carbon source [58]. LB medium is a much richer and more complex medium than GM and therefore sustains more rapid growth. We assumed that any resulting history-dependent differences in cell state at time t0 might lead to different history-dependent behaviors in the common medium after t0.

After experiencing one of the 10 different cell histories, cells were then pelleted and resuspended at an intermediate density (OD600 = 0.5) in a common stress medium, in this case, sporulation salts starvation medium (SM) [58]. The resuspension time is denoted t0. Thus, regardless of past experiences, all cells observed after t0 were subjected to starvation conditions starting at t0 in a fixed-density, fixed-size population.

Our three observables Y after t0 consisted of two fluorescent reporters, one for sporulation initiation and another for degradative enzyme synthesis (strain KEE (PspoIIE-gfp, PaprE-dsred cmp)), and optical density of the culture as a proxy for cell growth (OD600), measured at the bulk population level every 15 minutes for 24 hours starting at time t0 (see Fig. 5 for time series, and Materials and Methods for details on strain construction and experiments). Thus, with 30 cultures–three for each of the 10 cell histories– and three observables per culture measured every 15 minutes for 24 hours in the common stress medium starting at t0, the memory data compendium for this set of experiments consists of 30×3×96 = 8,640 measurements arranged in a 90 by 96 matrix.

Data analysis overview: The resulting memory data compendium was then analyzed for short- and long-term memory in each output signal individually and in all possible combinations of the three signals by applying the memory quantification algorithm described in detail in Materials and Methods and illustrated in the flow chart in Supplementary Information Section S2.

To briefly summarize, in order to estimate how much short-term and long-term memory was manifested in the behavior of the reporters, we sought to calculate the mutual information between the behavior of the cells after t0 and the history of the cells before t0. This calculation required that we estimate the joint probability density between cellular behavior after t0 and cell history prior to t0. Given constraints on the amount of data and other considerations described in detail in Section S3 of Supplementary Information, we took a clustering approach to this problem. That is, we first clustered the response of the pathway reporter as a way of dividing the trajectories into groups with common, distinct behaviors. The resulting assignment of each trajectory to a cluster was then used to calculate the frequency of co-occurrence of each behavioral class and each possible cell history. From this histogram we estimated the requisite joint probability distribution, which was then used to calculate the mutual information between cell history and the behavior of the observable, and thus arrive at an estimate for memory.

We performed this procedure on the 30 trajectories (3 replicates for each of the 10 cell histories tested) of each of the three observables, using both the short term (first 11 hours of measurements, during which the signal was still dynamically varying-see Materials and Methods for more details on our choice of analysis intervals) and long-term response (last three hours of measurements, from 21 to 24 hours, by which time the signals have remained flat for several hours) in order to estimate short-term and long-term memories manifested in each individual signal. To calculate the short-term and long-term memory in the combined activities of multiple signals, we took the same approach, with the one difference being that the clustering step captured the combined behavior of multiple readouts (Step 3 in the algorithm in Materials and Methods). All bit counts were then normalized to calculate memory fidelities and orthogonalities, as defined in Appendix S1, in order to estimate in relative terms how much of the total possible memory each system ‘remembers’, and how much ‘extra’ memory is embedded in the higher-dimensional spaces formed by multiple pathways.

Since the 30 populations were subjected to 10 different (within error) past conditions M = (Medium1, Density1) in equal proportions, the informational entropy of the cell history space M is H(M) = −log2(1/10) = 3.3219 bits. Thus, without prior knowledge there are 3.3219 bits of information about cell history at most that can be recovered from observation of these three outputs, either individually or in combination and on any time scale.

The B. subtilis stress responses measured by the three observables (Figure 5) appear neither memoryless nor in possession of a perfect memory of the cell histories tested. They do not appear to be memoryless because not all signals from a given observable follow a common trajectory (within noise bounds) irrespective of past history of the cells. Nor does the memory of any observable appear to be perfect, because though there are ten distinct cell histories prior to time t0, there appear to be fewer than ten distinct dynamics per observable in response to the starvation stressor administered at time t0. By eye, there appear to be more distinct behaviors in the short term than in the long term. Also, different cell histories group together for different observables. This means that we expect a higher bit count estimate of short term memory than long term memory, and different amounts of memory and of different aspects of cell history in the three pathway observables.