How basic components of information-updating interact to encourage variation in the results of empirical studies of within and transgenerational plasticity

Experiences of parents and/or offspring are often assumed to affect the development of trait values in offspring because they provide information about the external environment, but it is currently unclear how information from different sources and times might combine to affect the information-states that provide the foundation for the patterns observed in empirical studies of developmental plasticity in response to environmental cues. We analyze Bayesian models designed to mimic fully-factorial experimental studies of within and transgenerational plasticity (TWP), in which parents, offspring, neither or both are exposed to cues from predators, to determine how different durations of cue exposure for parents and offspring, the devaluation of information from parents or the degradation of information from parents would affect offspring estimates of environmental states related to risk of predation at the end of such experiments. We show that the effects of different cue durations, the devaluation of information from parents, and the degradation of information from parents on offspring estimates are all expected to vary as a function of interactions with two other key parameters of information-based models of TWP: parental priors and the relative cue reliability in the different treatments. Our results suggest empiricists should expect to observe considerable variation across experimental studies of TWP based on simple principles of information-updating, without needing to invoke additional assumptions about costs, tradeoffs, development constraints, the fitness consequences of different trait values, or other factors.


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The trait values expressed by the offspring of one generation can be affected by their own 34 experiences earlier in life (within-generational plasticity, WGP), by the experiences of their parents 3 3 35 (transgenerational plasticity, TGP), or by the combined effects of both (within and transgenerational 36 plasticity, TWP) [1][2][3][4][5]. It is often assumed that one reason why the experiences of parents and offspring 37 might have adaptive effects on offspring trait values is that those experiences provide information about 38 conditions in the external environment that the offspring are likely to experience later in life [6][7][8][9][10][11][12]. In 39 addition, information about conditions in the external environment can also be provided by genes, 40 inherited epigenic factors and parental phenotypes [7,13,14]. Hence, in order to appreciate how 41 information from an individual's distant and immediate ancestors and its own personal experiences 42 might combine to affect the development of traits that are the focus of studies of WGP, TGP or TWP, we 43 must consider how information from a variety of different sources combines within and across 44 generations to affect the information-state of that individual.

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In principle, Bayesian updating is the best way to combine information from different sources 46 and different times to estimate the value of variables in the external environment [15]. As a result, in recent years researchers have begun to use Bayesian approaches to study how information from 48 ancestors and personal experiences might combine over the course of ontogeny [16][17][18][19][20][21][22], review in [23].

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These models assume that an individual's estimate of conditions in the external environment can 50 change over ontogeny, as its initial estimate based on information from its ancestors (its naïve prior Hence, in the current study we considered how differences in the duration of exposure to the 150 same cues in parents and offspring and the devaluation or degradation of information from parents

Material and methods
156 Details about the design and biological rationale for the models used as the basis for those in 157 the current article are provided in previous publications [19,21,22,25]. In brief, we assumed that each 158 parent began with a prior distribution (Prior), based on information from their ancestors (e.g. via genes, 159 inherited epigenetic factors, grand-parental experiences), as well as any informative experiences the 160 parents had before the onset of the experiment. We illustrate the results for three informative parental 161 Prior distributions, with different means (0.1, 0.5 and 0.9), but the same variance (0.04) (For definitions 162 of these and other terms, see Supporting Information). 163 We assumed that cues from the predator in the P treatment were always informative, and used 164 beta functions with shapes modelled by α > β to indicate the shapes of the cumulative likelihood 165 functions for the conditions in the P treatments. We analyzed two sets of models which differed with 166 respect to their assumptions about the reliability of the information provided by the N treatment. In the 167 first set (N-models), the information provided by the N treatment was less reliable than the information 168 provided by the P treatment. We analyzed this situation by assuming that the information provided by 169 conditions in the P treatment was highly reliable (likelihood with a shape modelled by α = 8, β = 1) while 170 conditions in N provided no information about the state (i.e., the cumulative likelihood function for the 171 N treatment had a uniform distribution (α = 1, β = 1). In the second set of models (N* models), the 172 information provided by the P treatment and the information provided by the N treatment were equally Duration of exposure to the same cue in parents and offspring 188 Preliminary analyses showed that if parents and offspring were exposed to the same cues for 189 the same period of time, differences between the parental and the offspring generation in the age of 190 onset of the exposure period had no effects on the results. For instance, our analyses indicated that 191 offspring estimates of predator density would be the same if their parents had been exposed for two 192 weeks to cues from predators just before gamete production as if the offspring themselves had been 193 exposed for two weeks to the same cues as juveniles. These results occur because in Bayesian updating 194 models which assume that the true state of the environment is unlikely to change over time, if different 10 195 subjects with the same prior distribution are exposed to the same cues, the order in which they were 196 exposed to those cues has no effect on their final posterior distributions. Because the order-indifference 197 of Bayesian updating is particularly relevant to analyses of sensitive periods and age-dependent 198 plasticity, we defer discussion of this point to a study of that topic (Stamps, in prep.) 199 In contrast, preliminary analyses suggested that different durations of exposure to the same 200 cues in parents and offspring might have strong effects on offspring estimates at the end of their 201 respective treatment periods. In order to investigate the effects of different durations of exposure to 202 the conditions P or N for parents and offspring on the results, we divided the total treatment period, T, 203 for each generation into four intervals of equal length, and then specified the likelihood function for 204 exposure to the cue for one interval, and the likelihood function for no exposure to the cue for one 205 interval. For instance, to model a situation in which parents were exposed to the cues for a longer 206 period than the offspring, we assumed that parents in the P treatment were exposed to cues from the 207 predator for all four intervals, whereas the offspring in the P treatment were exposed to the cues from 208 the predator for either one, two or three intervals, and were exposed to no cues from predators for the 209 remaining interval(s). Similarly, to model a situation in which offspring were exposed to the cues for a 210 longer period than the parents, we assumed that offspring in the P treatment were exposed to the cues 211 for all four intervals, but parents in the P treatment were exposed to the cues for one, two or three 212 intervals, and were exposed to no cues for the remaining intervals. In all of these models, the N group 213 was maintained with no cues from predators for all four intervals. We then used Bayesian updating to As in [25], we computed separate models for the N-situation (in which the conditions during 218 one time interval in the P treatment provided much more reliable information than the conditions 219 during one interval in the N treatment) and for the N* situation (conditions during one interval in the P 220 treatment provided information as reliable as the conditions during one interval in the N treatment).

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For the N-models, we assumed that in a P treatment in which the individual was not exposed to the 222 cues in all four intervals, no information was provided in an interval that lacked the cue. That is, the 223 likelihood function for one interval spent in the absence of the cue had a shape indicated by a uniform 224 distribution (α =1, β = 1). In the N* models, we assumed that in a P treatment in which the individual 225 was not exposed to the cues in all four intervals, the information provided by one interval spent in the 226 absence of the cue was as reliable as the information provided by one interval spent in the presence of 227 that cue. For instance, if the likelihood function for one interval in the presence of cues from a predator 228 had a shape indicated by α = 2.5, β = 1,, the likelihood function for one interval in the absence of those 229 cues had a shape indicated by α =1, β = 2.5. it is assumed that the information provided by the signal that parents pass along to their offspring based 235 on their exposure to a given cue is less reliable than the information provided by the cue to which the 236 parents had been exposed. In Bayesian terms, this means that although we would expect the likelihood 237 function for the signal provided to their offspring by parents in the P treatment to have the same mean 238 value as the cue in the P treatment for the offspring, we would expect the likelihood function for the 239 parental signal to have a higher variance than the likelihood function for the cues in the P treatment for 12 240 the offspring. Hence, we used the same procedure to model the devaluation of information from the 241 parents and the degradation of information from the parents. That is, we assumed for both situations 242 that the signal from the parent and the cue for the offspring had likelihood functions with the same 243 mean, but different variances. For instance, instance, if the likelihood function for the P treatment for 244 offspring had a shape modelled by α = 8, β = 1 (mean = 0.89, variance = 0.01), the likelihood function for 245 the signal provided by parents to offspring by the parents in the P treatment might have a shape 246 modelled by α = 3.5, β = 0.44 (mean = 0.89, variance = 0.02). That is, we assumed that exposure to cues 247 from predators in the P treatment yielded the same point estimate of the value of the state of the 248 environment for parents and offspring (in this case, a relatively high value of 0.89, on a scale of 0 to 1),

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but the reliability of the information provided by the parents to their offspring (indicated by the variance 250 of the likelihood function) was lower than the reliability of the information provided by the same 251 experience for the offspring.

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In the N-models, the reliability of the information for the P treatments differed for parents and 253 offspring, as was indicated above. However, given our assumption for these models that the information 254 in the N treatments was unreliable (see above), we assumed that the information provided by the N 255 treatments was equally unreliable for both the parent and the offspring generation (modelled by α = 1, 256 β = 1).

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In the N* models, in which conditions in the N treatment provided information as reliable as 258 those in the P treatments, we devalued the information provided by parents in the N treatment groups 259 to their offspring. For instance, if the likelihood function for the P treatment for the offspring had a 260 shape modelled by α = 8, β = 1, the likelihood function for conditions in the N treatment for offspring 261 had a shape modelled by α = 1, β = 8, but the likelihood function for the information provided by 262 parents to their offspring had a shape modelled by α = 0.44, β = 3.5.

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13 263 All of the other assumptions and parameter values for these models were the same as those for 264 the models described in [25].

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In the 'baseline' models, in which parents and offspring were exposed to the same cues for the 269 same period of time, and in which information from parents was neither devalued or degraded, the to each other. However, this pattern only occurred when the state of the environment indicated by the 277 parental Prior differed from the state indicated by the cue in the P treatment. Since we assumed here 278 that the cues in the P treatment indicated that the value of the state of the environment was high, we 279 observed the jump-up pattern when the parental Prior strongly contradicted the cue (Prior mean = 280 0.1) (Fig 1a). In contrast, when the parental Prior and the cues in the P treatment indicated the same 281 value of the state (Prior mean = 0.9), the offspring estimates were similar for all four treatment groups.

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This result occurred because in these models, the N treatment had no effect on an individual's 283 information-state, so the results were primarily affected by the 'discrepancy rule' of Bayesian updating 284 (see [24] for more on this topic). 14 14 285 In the N* models, in which the information provided by conditions in the P treatment and the 286 information provided by conditions in the N treatment were equally reliable, we observed a 'step-up' 287 pattern, in which the offspring estimates for the NN treatment were low, those for the PP treatment 288 were high, and those for the NP and PN treatments were intermediate. In this case the pattern did not 289 vary as a function of the parental Prior: for the same set of parameter values, the step-up patterns were 290 similar for a range of parental Priors (Fig 1b). This result occurred because in the N* models, by the end 291 of the experiment reliable information about the state of the environment had been provided by the 292 cues in both treatment groups, so after the offspring were exposed to two doses of informative cues N treatment provide information as reliable as conditions in the P treatment; the likelihood functions for 310 the P and N treatments are mirror-images of one another. In this example, conditions in the P treatment 311 indicate with a high level of reliability that the state of the environment is likely to be high (likelihood 312 modelled by α = 8, β = 1); conditions in the N treatment indicates with equally high reliability that the 313 state of the environment is likely to be low (likelihood modelled by α =1, β = 8).

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When parents and offspring were exposed to the same cues for different periods of time, the 315 main effect was to generate differences between the offspring estimates for the PN and NP groups; 316 these differences were not observed in the baseline models (compare Fig 1 with Figs 2 and 3). As 317 intuition would suggest, the offspring estimates were higher for the treatment group which included the 318 generation that had been exposed to the cues for a longer time. For instance, if the offspring in the P 319 treatment group were exposed to the cues for a longer period than their parents, PN < NP (Fig 2a,b).

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Conversely, if the offspring in the P treatment group were exposed to the cues for a shorter time than 321 their parents, then PN > NP (Fig 3a,b). 322 323 Fig 2. Duration of exposure to the same cues is longer for offspring than for parents. In the P 324 treatment groups, parents are exposed to the presence of cues from predators for two of four time 325 intervals, but offspring are exposed to cues from predators for all four time intervals (see text). a. N-326 models. In this example, the likelihood function for exposure to cues from a predator for one time 327 interval has a shape indicated by α = 2.5, β = 1; the likelihood function for the absence of cues for one 328 interval has a shape indicated by a uniform distribution (α =1, β = 1). b. N* models. In this example, the 329 likelihood function for exposure to cues from a predator for one interval has a shape indicated by α = 330 2.5, β = 1; the likelihood function for the absence of cues for one interval has a shape indicated by α =1, 331 β = 2.5. Duration of exposure to the same cues is longer for parents than for offspring. In the P treatment 334 groups, parents are exposed to the presence of cues from predators for time four intervals, but offspring 335 are exposed to cues from predators for only two of the four time intervals (see text). Other variables are 336 the same as in Figure 2.

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The overall patterns were otherwise similar to those found in the baseline models. That is, a 339 jump-up pattern was detectable in the N-models when the parental Prior indicated a different value of 340 the state than the cues in the P treatment, but not when the parental Prior indicated a value of the state 341 similar to the value indicated by the cues in the P treatment (Figs 2a, 3a). Similarly, similar step-up 342 patterns were detectable across a range of parental Priors in the N* models (Figs 2b, 3b). However, for 343 comparable sets of parameter values (i.e. for the same parental Prior, and for the same likelihood 344 function for the cues in the P treatment), the differences between the offspring estimates for the NP 345 and PN groups were much less pronounced for the N-models than for the N* models (compare Fig 2a   346 versus 2b, and Fig 3a versus 3b).

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When information from the parents was devalued or degraded, the models predicted lower 348 estimates of the value of the state for the PN group than for the NP group (Fig 4). The overall patterns 349 were otherwise similar to those in the baseline models. A jump-up pattern was detectable in the N-350 models when the parental Prior indicated a different value of the state of the environment than did the 351 cues in the P treatment, but not when the parental Prior indicated a value similar to that indicated by 352 the cues in the P treatment (Fig 4a). In addition, similar step-up patterns were detectable for all parental 17 354 between the offspring estimates for PN and NP were much less pronounced for the N-models (Fig 4a) 355 than for the N* models (Fig 4b).
356 357 Fig 4. Information from parents is devalued or degraded before being passed to the offspring. We 358 assume that parents and offspring in the P groups are exposed to the same cue for the same period of devalued, relative to the information based on cue-exposure for offspring, and 5) the extent to which 377 information passed from parents to offspring is degraded, relative to the information based on the 378 personal experience of the offspring. We illustrate these findings by modelling fully factorial 379 experiments in which parents, offspring, both or neither are exposed to cues from predators (in P 380 treatments) or are not exposed to those cues (N treatments).

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Our results    capable of detecting the cues on their own. Hence, if both parents and offspring were continuously or 472 repeatedly exposed to the cues of interest from the embryo stage until just prior to offspring 473 production, both generations would be exposed to those cues throughout their sensitive periods, no 474 matter if or when they occurred in either generation. In that case, the predicted patterns of offspring 475 estimates would be the same as those predicted in the absence of sensitive periods (e.g. see

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Our results also confirm the intuitive notion that when information from parents is devalued 482 relative to information from offspring, exposure to cues for parents would have a weaker effect on 483 offspring estimates than equivalent exposure to the same cues in the offspring. Theory predicts that the 484 devaluation of information from parents should occur as an evolved, adaptive response to reduced 485 levels of autocorrelation between parental environments and offspring environments (see Introduction).

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In other words, the devaluation of information from parents is expected when, under natural conditions, 487 the value of a state of the environment is likely to change between the time that parents are exposed to 488 cues and the time that their offspring are exposed to the same cues. For instance, if we assume that the 489 true value of the state gradually changes over time, the devaluation of information from parents would 490 be inversely related to the amount of time that elapsed between the time that parents were exposed to 491 cues and the time that their offspring were exposed to the same cues. In turn, this implies that for 492 subjects from the same population, the devaluation of information from parents would be less 23 23 493 pronounced in an experiment in which parents were exposed to the cues of interest just before their 494 offspring were conceived and their offspring were exposed to the same cues soon after hatching than in 495 an experiment in which both parents and offspring were exposed to the cues soon after hatching. kairomones from a predator from birth to maturity in a P treatment is typically assumed to provide 583 reasonably reliable information about the density of that predator, while the absence of cues from the 584 predator from birth to maturity in an N treatment is assumed to provide comparably reliable 585 information indicating that the density of that predator is low. As predicted by the models described 2) Are the conditions in the different treatments likely to provide equally reliable information about a 597 state of the environment, or are the conditions in one treatment likely to provide much more reliable 598 information than those in the other treatment? As was shown here and in [25], we expect the patterns 599 of offspring estimates in TWP studies to dramatically differ in these two situations.

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3) Are parents and offspring exposed to the same cues for the same period of time? As is shown in the 601 current article, all else being equal, differences in the duration of cue-exposure for parents and offspring 602 are expected to affect the patterns of offspring estimates, especially if the cues in the different 603 treatments are equally reliable. 604 4) Is there strong existing support for the assumption that a particular response in a particular trait to a 605 particular cue is likely to be adaptive? In such cases, it is more likely that differences among the 606 treatment groups in offspring information states at the end of the experiment will be related to 607 differences among those groups in the trait values expressed by the offspring at the end of the 608 experiment.

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In addition, our results indicate that it may not be necessary to invoke assumptions about 610 developmental constraints, costs of sampling, or the fitness consequences of trait values of offspring to 28 28 611 account for at least some of variation in results observed in empirical studies of TWP. Instead, we have 612 shown that even if we restrict ourselves to experiments with information-only cues, we should expect to 613 observe considerable variation in their results, as a function of variation in parental priors, the reliability 614 of the cues in the different treatments, differences in the duration of cue-exposure for parents and 615 offspring, and the extent to which information from parents was either devalued or degraded in 616 comparison to information from offspring. 617 618