Negative feedback may suppress variation to improve collective foraging performance.

Social insect colonies use negative as well as positive feedback signals to regulate foraging behaviour. In ants and bees individual foragers have been observed to use negative pheromones or mechano-auditory signals to indicate that forage sources are not ideal, for example being unrewarded, crowded, or dangerous. Here we propose an additional function for negative feedback signals during foraging, variance reduction. We show that while on average populations will converge to desired distributions over forage patches both with and without negative feedback signals, in small populations negative feedback reduces variation around the target distribution compared to the use of positive feedback alone. Our results are independent of the nature of the target distribution, providing it can be achieved by foragers collecting only local information. Since robustness is a key aim for biological systems, and deviation from target foraging distributions may be costly, we argue that this could be a further important and hitherto overlooked reason that negative feedback signals are used by foraging social insects.

Introduction

Collectively-foraging social insects use feedback mechanisms in order to robustly and efficiently satisfy the nutritional requirements of the colony. Positive feedback signal usage by such foraging social insects is well known, such as mass-recruitment via pheromone in various ant species [1], and recruitment of small numbers of individuals such as via the honeybees’ waggle-dance [2], or rock ants’ tandem-running [3]. The use of negative feedback signals in these systems has, however received comparatively little attention. Negative feedback was predicted to be important for collectively foraging species [4, 5], and subsequently discovered in diverse systems such as Pharaoh’s ants [6, 7] and honeybees [8, 9]. Several studies have interpreted negative feedback as a mechanism to reduce recruitment to a resource based on some aspect of its quality, for example allowing unrewarded trails to be shut down [6, 7], allowing recruitment to a crowded source of forage to be reduced [10], or transferring information that a forage patch may have an increased predation risk [8, 11]. Subsequent studies have similarly focussed on the role of negative feedback in dealing with time-varying forage patches [12, 13], or with the amount of available comb storage space [14].

Here we propose an alternative function for negative feedback mechanisms in collective foraging, suppression of costly variation in the colony’s foraging performance. In the following, we present simple models of collective foraging with positive and negative feedback, and with positive feedback only. We show how both models are able to approach a desired target distribution over forage patches on average, when forager populations are assumed to be infinite. However, when finite forager populations are modelled, the two foraging systems differ in the robustness with which they achieve the target distribution; with positive feedback only, stochastic fluctuations can lead to the forager population being far from its target distribution at any point in time, however by adding negative feedback the forager distribution becomes more robust. We argue that this will increase colony-level foraging success [15, 16], and thus may represent a new functional explanation for the observation of negative feedback in foraging by social insect colonies.

Foraging theory is an active and complex research area, and our results do not rely on assumptions about the nature of the colony’s target distribution, other than it can be achieved by agents with access only to local information at both the forage source, and the colony. Thus, the target distribution may be akin to an Ideal Free Distribution, in which agents are distributed such that none can improve overall foraging efficiency by switching to a different forage patch [16, 17]. Alternatively, the target distribution may be based on the requirement of the colony for different micro- and macro-nutrients [18-21]. Or, the target distribution may be based on some other objective entirely, or on combinations of objectives such as those just discussed. In ignoring the nature of the distribution, therefore, our focus is purely on the dynamics of foraging, and how negative feedback can improve this.

For our analysis we adapt our model from a simple model of negative feedback for foraging in honeybee colonies [12], in itself inspired by models of negative feedback in house-hunting honeybee swarms [22-24]; however since other social insect species such as Pharaoh’s Ants also make use of negative feedback during foraging [6], we argue that the model is generally applicable.

Methods

We assume a target distribution of the individuals to the n patches in quantities proportional to the relative patch quality arbitrarily defined: where q is the quality of patch i. In our models an individual’s state can be either uncommitted (X) or committed to patch i (X) with i ∈ {1, …, n}. Therefore, based on the number of patches n, the commitment of the population will be split among n + 1 subpopulations; we represent the subpopulation proportions as x and x, in the closed interval [0, 1]. Note that, in a finite population of S individuals, it will be impossible for the colony to achieve exactly the desired target distribution if x S is not an integer number.

We analyse the population dynamics of the two systems parametrised to reach the same target distribution (with and without negative social feedback) using mean-field models of infinite and finite populations, using ordinary differential equations (ODEs) and stochastic simulation of the master equation respectively. Both types of analyses can be performed for models derived from chemical reaction equations, which specify how individuals in the system interact and change state (see Table 1).

Table 1

The two analysed models can be described in terms of transitions between commitment states by individuals.

The commitment states are ‘committed to foraging patch i’ (X) or ‘uncommitted’ (X). Both models have the same positive and negative feedback for independent transitions: quality-dependent discovery and constant abandonment (leak a). The difference lies in the social feedback; one model (blue) has quality-insensitive recruitment (r = ρ) but no negative social feedback (z = 0). The other model (red) has both quality-sensitive recruitment (r = ρq) and quality-insensitive self-inhibition (z > 0), as reported by field observations [30]. In these representative models, we set rates as constant and (linear) quality-sensitive functions of the quality according to the best function we obtain with numerical optimisation (see S1 Text).

		WITHOUT NEGATIVE SOCIAL FEEDBACK	WITH NEGATIVE SOCIAL FEEDBACK
Independent discovery	X_U→q_iX_i	Quality-sensitive	Quality-sensitive
Independent abandonment (leak)	X_i→aX_U	Constant	Constant
Recruitment (positive social feedback)	X_U+X_i→r_i2X_i	Constant	Quality-sensitive
Stop signalling (negative social feedback)	X_i+X_i→zX_U+X_i		Constant

The ODE model assumes an infinitely-large population size S and provides deterministic system dynamics in the absence of any noise from finite population effects. On the other hand, stochastic simulation of the master equation (Gillespie’s SSA [25]) gives a probabilistically correct simulation of dynamics of finite populations of size S.

While previous research has documented that collective foraging is regulated by the actions and interactions that we included in our models, the relationship between their frequency (transition rates) and the estimated nest-site quality are still debated. Table 1 reports the best functions we obtained through numerical optimisation to approximate the target distribution. Including negative feedback inevitably requires a change also in the recruitment function, from constant to linearly proportional to the quality. The difference in the recruitment function between the two models has not been imposed by design but is the result of the numerical optimisation analysis that we show in detail in S1 Text. Here, we assume that social recruitment (positive feedback) is much more efficient than independent discovery, so r ≫ q, as has been documented in a large variety of social insect species [26-29]. For fair comparison, the average recruitment strength r is equalised between the two models so that quality-sensitive recruitment transitions—model with negative feedback—happens on average at the same rate of quality-insensitive recruitment—model without negative feedback (see S2 Text). The model with only positive feedback is easy to solve for the desired equilibrium distribution of foragers, with a simple parameterisation of individuals’ rates (see S3 Text). The model with negative feedback, however, requires a heuristic individual parameterisation based on site qualities, which we perform numerically. However, this heuristic has a simple functional form (see Fig A in S2 Text) so could easily be approximated by real foragers.

Results

The two top panels of Fig 1 show the time dynamics of the two models for representative values and n = 3 patches. Both models asymptotically approximate the target distribution of Eq (1).

Fig 1

Temporal evolution of the models without (left) and with (right) negative social feedback in an environment with n = 3 food patches with qualities q1 = 0.75, q2 = 0.5, q3 = 0.25.

The top panels show the dynamics of the ODEs for systems of infinite size S → ∞. The bottom panels show the trajectories of 30 representative runs of the stochastic simulation algorithm (SSA, [25]) for a system comprised of S = 200 individuals. The boxplots on the right of each bottom panel show the statistical aggregate at time 400 for 1000 runs of the SSA. (Other simulation parameters are: constant abandonment a = 10−3, average recruitment strength r = 100, and stop signal strength z ≃ 3.1.) While the infinite size dynamics predict convergence to the target distribution of Eq (1) (dashed lines) for both models, the stochastic trajectories show different results for the two models. The system without negative social feedback has smaller fluctuations over time but frequently stabilises at values far from the target distribution (bottom-left panel). The system with negative social feedback fluctuates more but always remains relatively close to the target distribution (bottom-right panel). The apparently quicker dynamics of the ODE model for the system without negative social feedback are due to the symmetric initial conditions. In the left inset, we show that a small perturbation of the initial population (i.e. x1, x2 = 0 and x3 = 0.05) delays the convergence by more than 5 orders of magnitude. Such a susceptibility to random fluctuations is made evident by the stochastic trajectories. The right inset shows a zoom of the larger plot.

Through numerical integration of the master equations, we investigate the effect of stochastic fluctuations on the system dynamics [25]. The fluctuation size is inversely proportional to the system size S, i.e. there are no fluctuations in very large groups (i.e. S → ∞) and large fluctuation in small groups. The effect of the system-size noise can be appreciated in the two bottom panels of Fig 1. They show 30 representative trajectories for a system of size S = 200. The higher variance can also be appreciated in the boxplots on the right of each bottom panel of Fig 1, in which the average of 1,000 simulations hits the target value in both models; however, the variance is reduced considerably with the introduction of negative social feedback. These results are not specific to the representative example of Fig 1, but are consistent throughout the wide parameter space (see analysis in S4 Text). Additionally, increasing abandonment, which is a form of independent, asocial negative feedback, is not sufficient to reduce variance (see S5 Text).

Large deviations from the target distribution could compromise the ability of the colony to intake the necessary nutrients for survival and reproduction, thus decreasing colony fitness. Fig 2 shows how the error in achieving the target distribution is significantly higher without negative social feedback. Similarly, the speed of adaptation to environmental changes is an important factor in the survival of the colony [31, 32]. The system without negative feedback can be incapable of adapting to changes in a timely manner because its temporal dynamics vary significantly depending on the initial commitment (see top-left inset of Fig 1). The system with negative feedback, instead, displays a constant convergence time regardless of the initial state of the system (see S6 Text). Fig 3 shows how the convergence speed and the deviation from the target distribution are influenced by the strength of the negative feedback; the strength of negative feedback can tune a speed-robustness trade-off, similarly to the tuning of speed-value and speed-coherence trade-offs in consensus decisions [23, 24, 33]. In agreement with field observations of honeybees, which increase stop signalling when a quick response is necessary [10], our analysis also predicts a speed-up of the group dynamics for higher levels of negative feedback. The dynamics of Fig 3 can also be discussed in light of the results of a recent empirical study which has found that sensitivity to inhibitory signalling changes as a function of colony size [34]. Our model, by linking the negative feedback rate with convergence speed and the deviation from the target distribution, offers predictions that can be tested on colonies of different sizes. For example, the time to adaptation to environmental changes is expected to be longer in smaller colonies, where bees show a lower sensitivity to inhibitory signalling [34], than in larger colonies.

Fig 2

Sum of squared errors (SSE) computed as the sum for n = 2 food patches of the square of the difference between the subpopulation size at time 1000 (convergence) and the target distribution to that patch (see S7 Text).

The boxplots show the distribution of the SSE for 103 numerical simulations for swarm size S = 200, average recruitment strength r = 100, and qualities q1 = 0.75 and q2 = 0.5.

Fig 3

The stop signalling strength can be the control parameter in a speed-robustness trade-off.

Stronger stop signalling speeds up the convergence of the system (magenta curve) but also increases the predicted error from the target distribution (blue curve). These results are in agreement with field observations that documented an increase in stop signalling when a quick response to environmental changes was necessary [10]. The inset shows that for increasing recruitment strength ρ both convergence time and error decrease. Both error and convergence time are computed from the infinite population model (ODE). The error is computed as the sum for every foraging population of the squared distance R2 from the target at large time (convergence, computed analytically as the ODE’s stable fixed point in the unit-simplex). The convergence time (magenta curve) is computed as the time necessary to reach the (numerically computed) fixed point. As the system has an asymptotic convergence, the reported time corresponds to the R2 error becoming smaller than 10−4.

Discussion

Negative feedback has been considered in collective decisions, particularly as a means of symmetry breaking [22–24, 35], and in foraging, as a means of adapting to dynamically changing environments [7, 10, 12, 13]. Other than in entomology, negative feedback has been observed as a tool for noise reduction in gene networks [36-38] and in electronic systems [39, 40]. Here we have shown that negative feedback may play an important role in reducing variance in colony foraging performance. For example, considering the honeybee system that inspired our model, field observations have reported that levels of stop signalling increase in response to changes such as dangerous, overcrowded, or depleted food patches [10, 11, 13, 41]; however, it has not yet been fully understood why, even in static conditions, honeybees always deliver a small number of stop signals to foragers visiting the same forage patch [10, 13, 34]. This pattern is consistent with our model, and the analysis presented is an interpretation for such observed behaviour.

Our results suggest a further progression in the evolution of collective foraging behaviour; solitary foraging by members of social insect colonies evolved first, but was comparatively inefficient due to the need for foragers to repeatedly and independently discover forage sites [42, 43] (see S8 Text). Subsequently, positive social feedback evolved to improve foraging efficiency [44-46], but this came at the expense of robustness of the foraging outcome, through increased variance in foraging performance (see S9 Text). Finally, negative feedback evolved not only to respond better to changing environments, but also to reduce variance in foraging performance. The re-use of negative feedback signals, such as in the case of honeybee stop-signals which are used in both foraging [10, 34] and house-hunting [22] life history stages, would facilitate performance-enhancing innovations in signalling behaviours; however, it is not clear whether stop-signalling first arose in foraging or in house-hunting contexts (intuitively, we suggest the former, a more common life history event).

Some species have not evolved negative signalling mechanisms but rely on natural decay of feedback, such as pheromone evaporation. For instance, Lasius niger ants rely on the downregulation of positive feedback (i.e. pheromone deposition) in order to let pheromone decay take over [47, 48]. It is worth noting that this is not technically negative feedback; given the time taken from the first observations of the positive-feedback signals in colonies of honeybees and ants [3, 49, 50] to that of the corresponding negative feedback signals [6, 51], it may be worth further exploring social insects in which explicit negative feedback has not been observed, to search for expected negative feedback mechanisms, or explain why their life history means they would not be beneficial. As a motivating example, decreases in the number of waggle circuit repetitions in honeybee swarms were taken to be due to decay processes internal to scout bees [52], but the negative stop-signal was subsequently discovered to be significant in these swarms [22].

We conclude by noting that our study highlights the importance of using multiscale modelling to understand collective behaviour [53-55]. In fact, through mean-field analysis we could not observe the dynamics that justify the use of the negative feedback. Instead, complementing the analysis with probabilistic models, we have been able to identify the system dynamics that favour the appearance of stop signalling as a mechanism for variance reduction. Multiscale modelling is a valuable framework which combines the use of a set of modelling techniques to analyse the system at various levels of complexity and noise. In this study, we only employed noise-free mean-field analysis and master equations with system-size dependent noise. However, further analysis could include the impact of spatial noise, and time-correlated information and/or interactions [53].

Relationship between transition rates and food patch quality.

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Parameters of the models.

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Stability analysis.

(PDF)

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Variance reduction in a large parameter space.

(PDF)

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Effect of asocial negative feedback (abandonment or leak α).

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Adapting to changing conditions.

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Sum of squared error.

(PDF)

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Asocial model.

(PDF)

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Large deviation from target.

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8 Mar 2022

Dear Dr Reina,

Thank you very much for submitting your manuscript "Negative feedback may suppress variation to improve collective foraging performance" for consideration at PLOS Computational Biology. As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. The reviewers appreciated the attention to an important topic. Based on the reviews, we are likely to accept this manuscript for publication, providing that you modify the manuscript according to the review recommendations.

The reviewers all agree that the presented results are valuable and worth publishing. In addition, they make a number of constructive suggestions to further increase the clarity of the paper. In particular, they suggest to

(-) discuss, in the main text, the exact differences between the two models. In particular, it would be useful to have a detailed discussion why recruitment is labeled as constant in the model without negative social feedback, but labeled as quality-sensitive in the model with negative social feedback.

(-) provide more detail about the mechanism that drives the observed reduction in variance, and

(-) move some of the supplementary material to the main text.

I concur with all these suggestions, and I'd like to encourage the authors to make the corresponding changes.

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The reviewers all agree that the presented results are valuable and worth publishing. In addition, they make a number of constructive suggestions to further increase the clarity of the paper. In particular, they suggest to

(-) discuss, in the main text, the exact differences between the two models. In particular, it would be useful to have a detailed discussion why recruitment is labeled as constant in the model without negative social feedback, but labeled as quality-sensitive in the model with negative social feedback.

(-) provide more detail about the mechanism that drives the observed reduction in variance, and

(-) move some of the supplementary material to the main text.

I concur with all these suggestions, and I'd like to encourage the authors to make the corresponding changes.

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: This paper describes a carefully thought out model of collective foraging in social insect colonies, and shows a new potential role for negative signals within collective foraging processes. While such signals had previously been assumed to be beneficial due to their role in directly communicating a negative state (lack of food, danger etc), this model tests the idea that such signals could also make forager allocation more robust, especially in limited foraging populations, as social insects frequently have.

The model description is very well laid out. The logic underlying the assumptions and decisions is clearly explained. The supplementary material is likewise clear, and support the main results well.

The duel modelling approaches are a particular strength of the paper, and the differences in the outcomes between the two modelling approaches are clearlydemonstrated and explained.

I did find table 1 a bit confusing, because at first reading, it seemed to suggest that the negative-feedback model also has an additional change to the positive feedback process, thus confounding the effect due to negative feedback. After careful reading of the text, I see that this is not the case, but the terminology is a bit non-intuitive. It would be possible to have quality-dependent positive-only recruitment, and I think that some small changes to the section of the text that justifies the introduction of quality dependence (lines 87-90) and/or to table 1 could make things clearer in this regard.

To summarise, I have no substantive concerns about this manuscript, and consider it to provide novel insights to the field of collective decision making.

Reviewer #2: This manuscript provides an interesting and important new potential function for inhibitory signals in social insect collectives. By using the example of the honey bee stop signal, it shows, with modeling, that adding negative feedback allows colonies with a realistic number of foragers to arrive at the correct foraging distributions. This is interesting because assuming an infinite colony size results in correct distributions, with and without negative social feedback, but at realistic colony sizes, the colony can arrive at decisions that are not optimal, based upon stochastic initial conditions. Further, errors in achieving the correct distribution are significantly higher without such negative feedback.

I think the manuscript is well written and the results clearly explained. Although I cannot comment on the details of the modeling, the results make sense given what is known about how collectives behave in response to such negative feedback.

One suggestion is that the authors consider the recent paper by Bell et al. (2021) showing that response to stop signals changes as a function of colony size and incorporate this in their discussion. It would be interesting to consider this paper in light of Fig. 3 given that the response of colonies of different sizes are, in a way, similar to considering the different stop signaling strength levels shown in this figure.

Bell, Heather C., et al. "Responsiveness to inhibitory signals changes as a function of colony size in honeybees (Apis mellifera)." Journal of the Royal Society Interface 18.184 (2021): 20210570.

With respect to Fig. 3, I would welcome more discussion or explanation. It seems that the optimal stop signal strength occurs at z=0.4 but that increasing z results in a very large increase in predicted error. Is this a result of using the infinite population model? Would using a finite, more realistic population model yield lower predicted errors as z increases? Also, the inset figure with recruitment strength on the x-axis should be mentioned in this legend.

In terms of the text, I have a small suggestion:

L155. I suggest, “As a motivating example, decreases in the number of waggle circuit repetitions in honeybee swarms” to clarify that the waggle phase durations are not decreasing, jut the number of waggle dance repetitions.

Reviewer #3: The authors identify an interesting and potentially generalizable finding about the role of negative feedback in collective decisions. Namely, they have developed a stochastic dynamical model of collective foraging that involves the key behaviors of discovery, abandonment, recruitment, and stop-signaling (the primary negative feedback interaction). The employ both a mean field model and a fully stochastic model to examine the relative contribution of negative feedback to the speed of approach to and relative value of the quasiequilibria of the stochastic system. Both a model without and a model with negative feedback exhibit steady states aligned or roughly aligned to an ideal free distribution in the mean field limit. However, when finite size effects are considered, the model without negative feedback does not reliably approach the desired distribution. Lastly, it is found that mild deviations in the eventual distribution increase for stronger negative feedback, so there is an optimum level of feedback that best provides reduced convergence time and relatively low error.

The paper is short, but the main findings are apparent in the results section and figures. What is not exactly clear to me is how the model itself brings about the reduction in variance, even in the supplemental information. Perhaps this is primarily a simulation-based result, but it seems like the authors should be able to explain in more detail precisely how the variance-reduction mechanism works. My intuition would tell me that perhaps the additional term is somehow increasing the stability (increasing the amplitude of the main negative eigenvalue) associated with the steady state. Is this something the authors could show for the full stochastic system? Is this something that would appear in a system-size expansion approximation of the full stochastic system? Seems like there should be some analysis that could be done to make it clearer. I would appreciate any response the authors could give to these queries in a revision.

Other than this, I would say that the authors could probably move some of the results from the Supplementary Info document up to the main text. It's written rather like a PRL or PNAS article with a lot of interesting analysis tucked into the supplement, but I'd say the sections ST1 and ST3 and their figures could be moved up to the main text.

The model is not particularly novel by any means and it certainly has been analyzed by these authors and others a lot in previous works, but nevertheless this study sheds light on a newly observed dynamical phenomenon of the model. The authors properly frame the findings and provide a nice set of analyses. As mentioned above, the paper would primarily benefit from a clearer and more detailed explanation of exactly (in terms of a dynamical systems or stochastic processes analysis if possible) how negative feedback brings about variance reduction. If the authors could add more detail on this, I would be happy to consider a revision.

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Reviewer #1: Yes

Reviewer #2: Yes

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24 Mar 2022

Submitted filename: Reply_to_reviewers.pdf

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8 Apr 2022

Dear Dr Reina,

We are pleased to inform you that your manuscript 'Negative feedback may suppress variation to improve collective foraging performance' has been provisionally accepted for publication in PLOS Computational Biology.

Before your manuscript can be formally accepted you will need to complete some formatting changes, which you will receive in a follow up email. A member of our team will be in touch with a set of requests.

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The manuscript has been sent to two of the reviewers who handled the original manuscript.

Both reviewers are happy with the changes, and so am I.

(I am personally leaning with reviewer #3 about moving more of the supplementary material into the main text; however, if the authors prefer to keep their manuscript as is, that's fine for me).

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #2: The authors have addressed my comments and I believe that the manuscript is now ready for publication.

Reviewer #3: Thanks to the authors for considering my comments and pursuing additional analysis to test the origins of the variance-reduction mechanism. The response makes good sense, and I think the resulting update to the paper is nice. Concerning moving supplementary up to the main text, I guess that's my bias as an applied mathematician, and I could see that for biologists this could be too much. The paper in its current form is in good shape for publication in PLoS Comput. Biol. as far as I'm concerned.

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Reviewer #3: Yes

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11 May 2022

PCOMPBIOL-D-22-00055R1

Negative feedback may suppress variation to improve collective foraging performance

Dear Dr Reina,

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With kind regards,

Livia Horvath

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