A mathematical model of non-photochemical quenching to study short-term light memory in plants

Plants are permanently exposed to rapidly changing environments, therefore it is evident that they had to evolve mechanisms enabling them to dynamically adapt to such fluctuations. Here we study how plants can be trained to enhance their photoprotection and elaborate on the concept of the short-term illumination memory in Arabidopsis thaliana. By monitoring fluorescence emission dynamics we systematically observe the extent of non-photochemical quenching (NPQ) after previous light exposure to recognise and quantify the memory effect. We propose a simplified mathematical model of photosynthesis that includes the key components required for NPQ activation, which allows us to quantify the contribution to photoprotection by those components. Due to its reduced complexity, our model can be easily applied to study similar behavioural changes in other species, which we demonstrate by adapting it to the shadow-tolerant plant Epipremnum aureum. Our results indicate that a basic mechanism of short-term light memory is preserved. The slow component, accumulation of zeaxanthin, accounts for the amount of memory remaining after relaxation in darkness, while the fast one, antenna protonation, increases quenching efficiency. With our combined theoretical and experimental approach we provide a unifying framework describing common principles of key photoprotective mechanisms across species in general, mathematical terms.

(Zx) in energy dissipation. Related to the qE mechanism, 23 a synergistic action of Zx and the thylakoid lumen pH has 24 been proposed [16], explaining why highest qE levels are 25 inducible only in presence of Zx [4,17,18]. 26 Based on titration studies under in vitro conditions, 27 Horton and co-workers suggested that Zx shifts the pH- 28 dependence of qE by about 1.5 pH units towards higher 29 lumen pH [18,19]. Furthermore, Zx was shown to modu-30 late the kinetics of NPQ induction (faster in presence of 31 Zx) and relaxation (slower in presence of Zx) [4, 20, 21], 32 and was proposed to accelerate the reorganisation / ag-33 gregation of the PSII antenna [22] that accompanies 34 qE [16,23,24,25,26]. These characteristics led to the 35 development of the 4 state model of qE [16,25,27], which 36 consistently explains the modulation of qE by the lumen 37 pH and Zx, irrespective of the underlying quenching mech-38 anisms and quenching sites involved in qE [24,28]. 39 Moreover, correlations of the Zx reconversion kinet- 40 ics with the relaxation of the slower NPQ components 41 qZ and qI indicate a function of Zx also in these pro-42 cesses [4,29,30,31,32,33,34], and Zx reconversion can 43 be considerably down-regulated or inhibited under stress 44 [35,36] and photoinhibiting [36,37] conditions. This grad-45 ual down-regulation of Zx epoxidation in response to dif-46 ferent light stress conditions thus allows the light stress-47 specific adjustment of the time of Zx accumulation, and 48 hence the modulation of the NPQ response in dependence 49 Using PAM chlorophyll fluorescence analysis we quantita-111 tively investigated the effect of short-term light memory 112 on NPQ by comparing the fluorescence patterns of the 113 first and second light periods. In the present study (see 114 Fig. 1A), we examined two factors affecting the light mem-115 ory in plants: intensity of incident light (varying from 100 116 to 900 µEm −2 s −1 ) and the relaxation time between first 117 and second light exposure (15, 30 or 60 min). 118 We first analysed whether the quenching patterns differ 119 between the two phases of light. We directly analysed the 120 originally measured maximal fluorescence (F ′ M (t)) data in-121 stead of derived NPQ values (visualised in Fig. S3), to 122 avoid mathematical distortion of the kinetics and provide 123 more reliable information on the mechanism [45]. Fluores-124 cence measurements are a non-invasive method for moni-125 toring photosynthetic dynamics, providing information on 126 the photosynthetic efficiency, protection and energy dis-127 sipation. However, each measurement can only be rela-128 tive [46], and therefore at first data is normalised to the 129 maximal fluorescence (measured after the first saturating 130 pulse of light applied to a dark adapted plant, F M ) and 131 then averages and standard deviations of the three repli-132 cates are calculated. In Fig. 1B we visualise the maxi-133 mal fluorescence kinetics in the first (training) and second 134 (memory) light phase (shifted to time 0). To highlight the 135 key features which we aimed to explain with the mathe-136 matical model, only the last two measurements taken in 137 dark phase (marked by grey background) and the first 138 five measurements taken in consecutive light phases are 139 displayed (for full traces of all 22 points taken during the 140 experiment see either the NPQ trace in Fig. S3 in the Sup-141 porting Information (SI Text) or extract the data from the 142 database).

143
It can be observed that for all light intensities the last 144 F ′ M in the relaxation phase (denoted F M ⋆ ) is consistently 145 lower than in the training phase (F M ). Likewise, the first 146 measurement in light at 61 s shows lower fluorescence than 147 the corresponding point in the training phase (see Tab. S1 148 for statistical significance). This timed response to pre-149 viously experienced illumination clearly demonstrates a 150 short-term memory. The extent of the incomplete relax-151 ation is influenced both by light intensity and the time 152 spent in darkness (see Fig. S4).

153
Based on our current understanding, these observations 154 can be attributed to the dynamical changes in the pig-155 ment composition, especially the slow epoxidation of zeax-156 anthin to violaxanthin in darkness. To quantify the zeax-157 anthin contribution to the memory effect we measured the 158 pigment composition at the end of each phase of the ex-159 periment (full analysis summarised on Fig. S5). Fig. 2 160 shows that after exposing the samples for 15 minutes to 161 high light intensities, Zx levels significantly increased up 162 to 50% of all xanthophyll cycle pigments (sum of violax-163 anthin, antheraxanthin (Ax) and zeaxanthin). Simultane-164 ously, one hour in dark was sufficient to reduce this by 165 half, explaining lower quenching effects in samples kept 166 for longer periods in dark. This decrease was not as pro-167 nounced under illumination with the lowest light intensity. 168 Moreover, zeaxanthin concentrations alone cannot explain 169 Figure 1: (A) Design of the experiment. Dark adapted plants were exposed to a first saturating pulse of light (SP) from which physiological parameters such as maximal fluorescence (F M ) and photosynthetic yield (ΦPSII) were derived. 30 s into darkness a second SP was applied, then the light was switched on for a fixed (training) period of 14 minutes. SPs were applied in a defined sequence (see Fig. S1), yielding maximal fluorescence (F ′ M ). The training period was followed by a dark period, interrupted by six SP to follow the fluorescence relaxation dynamics. Subsequently, in the second (memory) period, the same illumination intensity as in the training period was applied. Each experiment was repeated three times for three light intensities (100, 300 and 900 µEm −2 s −1 ) and different relaxation times (15, 30 and 60 min). (B) Comparison of the first two F ′ M measurements taken in darkness and first five measurements in training phase (blue points) with the last two measurements taken in the relaxation phase and first five measurements in the second light phase (red points). The first measurement taken in the light (at 61 s) is lower in the memory phase than in the training phase, regardless of light intensity and time of relaxation. Error bars indicate standard deviations for three replicates except for the experiment marked with *, where the experiment was repeated eight times (see the SI Text for motivation).   Because a number of applied simplifications (see SI Text 208 for justification and details) may raise concern whether the 209 system will exhibit a biologically meaningful steady state

236
These considerations lead to the overall equation for the 237 quencher activity: where [Zx]+kZSat reflects the contribution of Zx to the 239 quencher and k ZSat is a half-saturation constant. The over-240 all concentration of xanthophylls ([Vx] + [Zx]) is assumed 241 to be constant and the temporal changes of [Vx] and 242 [PsbS] are determined by appropriate differential equa-243 tions (described in the SI Text, Eq. 9 and 10). Thus, 244 individual and combined effects of both players on the 245 quenching dynamics could be quantified.

246
The γ parameters were fitted to the fluorescence traces 247 and their effect on the steady state and quencher activity 248 was extensively studied using metabolic control analysis 249 [52] (Fig. S7). The parameter γ 0 describes the baseline 250 quenching that does not require activation (and therefore 251 is also active in a dark adapted sample) and was included 252 to account for a small quenching observed in double mu-253 tants, where both PsbS and zeaxanthin dependent activa-254 tion is removed [4]. The parameter was fitted to reproduce 255 the double mutant behaviour.
where H(x) is the Heaviside function and PFD stands for 284 photon flux density, expressed in µEm −2 s −1 .
where k H · Q is the rate of NPQ, modulated by the in the simulation than in the experiment. Nevertheless, 314 we managed to reproduce the qualitative drop between 315 the corresponding points, as simulated F ′ M in the mem-316 ory phase is still lower than the corresponding simulated 317 F ′ M in the training phase. The steady-state fluorescence 318 values F s show discrepancies especially in the low and in-319 termediate light intensities, but are very well captured for 320 high light and in dark periods. One possible explanation 321 for the deviations of model results from experimental data 322 is that our model does not include other photoprotective 323 mechanisms that may affect F s , in particular state transi-324 tions [57,58,59]. To study this, a more complex model is 325 needed that is specifically designed to investigate the cross-326 talk and interplay between these two acclimation mecha-327 nisms. Since in the present study the focus lies on energy-328 dependent quenching, we consider the agreement of model 329 and experiment as satisfactory and decide to sacrifice some 330 degree of precision for the sake of a simple model structure. 331 A clear benefit of computer simulations is the possibil-332 ity of following the dynamics of otherwise hard to mea-333 sure molecules. In  Fig. 5B we provide the 346 same information in a phase phase plot, where the pH is 347 displayed as a function of quencher activity. This represen-348 tation clearly indicates the different timescales on which 349 the system operates. Trajectories start in the dark state 350 (low Q, pH close to 8) by rapidly reducing pH (vertical 351 drop), before the quencher is activated (curved trajecto-352 ries), and eventually the steady-state in constant illumi-353 nation (red dots) is reached. The memory is visualised 354 by the fact that the trajectories do not revert back to the 355 initial dark state after the first dark relaxation phase, and 356 thus the trajectories during the memory phase differ from 357 those of the training phase.

358
With the presented mathematical model we are able to 359 simulate different fluorescence kinetics in the training and 360 memory phase, moreover we are able to explore which in-361 ternal variables are correlated with this different dynamics 362 and quantify their contribution.  364 It is a well recognised issue in the field of systems biology 365 that modelling biological processes often requires acquir-366 ing a number of parameters [60] of which some might have 367 a physical meaning, some may be a rough approximation 368 of measured values, others may be fitted and some may 369 be simply impossible to measure with current techniques. 370 Much focus was put on developing optimization algorithms 371 that will ease parameter estimation, but the only safe way 372  to reduce the risk of over-fitting is to minimise the dimen-373 sions of the parameter space. One of the difficulties of 374 using available and published kinetic models is their often 375 huge parameter space, which makes them hard to adapt to 376 study analogous mechanisms in other species. To demon-377 strate that our model is of sufficient simplicity to allow for 378 an easy adaptation to a new and not extensively studied 379 organism, we have adapted our model to the ornamental, 380 shadow-tolerant plant Epipremnum aureum, also referred 381 to as Pothos. This choice was motivated by the finding 382 that shade-tolerant plants are characterised by longer last-383 ing memory for leaf illumination, as compared to plants 384 found in semi-arid climates [61]. 385 We therefore collected the fluorescence data for Pothos 386 using the same experimental setup as described above for 387 Arabidopsis (see Figs. S1 and S2). The analysis of the 388 NPQ dynamics demonstrated that under identical condi-389 tions this plant exceeds the quenching capacity of Ara-390 bidopsis (Fig. S3) confirming similar observations on other 391 shadow-tolerant plants [61]. In fact, under moderate light 392 Pothos already behaves like Arabidopsis exposed to high 393 light. With no additional information available, we as-394 sumed that the basic principle of the protecting mecha-395 nism is the same in the two plant species, but since Pothos 396 exhibits higher quenching capacity, we can reflect it by in-397 creasing the parameter γ 2 . We measured the chlorophyll 398 content in both species (Tab. S2) and found a 70% higher 399 content in Pothos than in Arabidopsis. With limited infor-400 mation on the electron transport chain protein abundance, 401 we kept the same values of all internal parameters as for 402 Arabidopsis and explained the more sensitive quenching 403 response to light by increasing the factor converting pho-404 ton flux density to light activation rate. With only those 405 two changes in the parameter space we reproduced the 406 experimentally observed fluorescence and photosynthetic 407 yield kinetics for Pothos (Fig. 6).

408
The agreement between the simulation and experiment 409 allows us to suggest that the enhanced quenching capac-410 ity in Pothos can be explained by a more efficient energy 411 transfer from the chlorophylls to the quencher. Possible 412 quenching sites or a closer spatial arrangement. To exper-  is remarkable that these findings could be obtained by a 433 very simple model structure. 434 We realise that very detailed models have the poten-435 tial advantage that they provide a very specific and ac- features of the investigated system which give rise to its 444 characteristic emergent properties. Therefore, developing 445 a simple model that can explain the same or even more 446 than a more complex model built for the same or simi-447 lar purpose, is in itself a useful scientific exercise, which 448 generates a deeper understanding of the biological sys-449 tem. Moreover, simple models are easier to implement 450 and can be generalised, but often are limited in the quan-451 titative reproduction of the data. For instance, a model 452 of high-energy state quenching of extreme reduced com-453 plexity aims at reproducing the key biological features of 454 quenching [56] but is clearly limited in its capability to 455 correctly reproduce the dynamics of quenching induction 456 and relaxation simultaneously. On the other hand, the de-457 tailed computer model of C3 photosynthesis by Laisk et 458 al. [62] was built to test whether the current understand-459 ing of photosynthesis is basically correct and is able to cor-460 rectly reproduce steady state behaviour of photosynthesis 461 and carbon fixation, but cannot reproduce NPQ kinetics. 462 A recent model that specifically aimed at understanding 463 the transient dynamics of NPQ, was proposed by Zaks et 464 al.
[63]. Employing a set of 26 nonlinear differential equa-465 tions, but only considering one quenching state, the model 466 is able to reproduce the quantitative difference in the fluo-467 rescence yield between low and high light conditions. The 468 complexity of the latter two models makes it difficult to 469 derive general conclusions that may be valid beyond the 470 boundaries of single species. Further, their size makes a 471 de novo implementation a very time-consuming task and 472 the reproduction and verification of the model results is 473 tedious, in particular since they are not provided as open 474 source code. 475 We have therefore presented here a new, highly simpli-476 fied mathematical model of NPQ, and employed the model 477 not only to accurately describe the rapidly reversible com-478 ponents of non-photochemical quenching, such as the pre-479 viously published models [56, 63], but further used it to 480 explain the phenomenon of short-term light memory, and 481 moreover provide a quantitative understanding of the dif-482 ferent contributions of the well-known NPQ components. 483 Our model accurately simulates the changes in the fluo-484 rescence yield at low, moderate and high light intensities 485 (Fig. 4). It further provides an explanation for the higher 486 extent of quenching observed for plants which have pre-487 viously been illuminated. Furthermore, it supports the 488 notion that the same organisational principles of photo-489 protective mechanisms are present in plants as different 490 as Pothos and Arabidopsis.

491
With a simple experimental setup with two light expo-492 sure periods separated by a varying relaxation time, we 493 demonstrate that the extent of quenching does indeed de-494 pend on how long ago and how much light a plant has 495 previously experienced -a behaviour which can safely be 496 termed memory. We could demonstrate experimentally 497 and theoretically that this light memory can be attributed 498 to the slow quenching component associated with the de-499 epoxidation of violaxanthin to zeaxanthin triggered by low 500 lumenal pH, which is in agreement with our current knowl-501 edge on NPQ and memory [16,25,27]. In the dark, epox-502 idation of zeaxanthin to violaxanthin is slow, so that even 503 after 30-60 minutes the conversion is not complete. In a 504 second exposure to light, the rapidly protonated antennae   568 We have demonstrated that our current understanding of 569 quenching processes can be converted into more general, 570 mathematical terms and with the implemented theory we 571 can reproduce the most critical behavioural features of 572 short-term illumination memory. This memory is gener-573 ated by the interaction of two components of NPQ, that 574 were previously identified by many others. The slower one, 575 accumulation of zeaxanthin, accounts for the amount of 576 memory lasting after relaxation in darkness, while the fast 577 one increases the efficiency of quenching. However, our 578 experiments do not provide evidence for an acceleration 579 of quenching activity by previous light exposure. Rather, 580 we propose to explain the consistently lower F ′ M in the 581 first seconds of the second light period by accumulation of 582 Zx only. Therefore, plants with active short-term mem-583 ory of previously experienced light initiate their photo-584 protection with some head-start, but at the same speed. 585 Moreover, our computational model supports hypotheses 586 on why shadow-tolerant plants exhibit a higher quenching 587 capacity. Together with this manuscript we provide all 588 necessary files to repeat and perform further experiments 589 in silico, therefore we encourage our readers to treat this 590 adaptation as an example of how our model can be used 591 to test hypotheses regarding NPQ in other, also less stud-592 ied organisms. Further, because of its simplicity and easy 593 adaptability, the model has the potential to support knowl-594 edge transfer from lab to field. Especially in combination 595 with cheap and easy-to-use devices to measure photosyn-596 thetic parameters outdoors (such as MultispeQ designed 597 by the PhotosynQ project [66]), our model provides a the-598 oretical framework in which the multitude of data can be 599 interpreted in a sophisticated way. Thus, it can serve as 600 a bridge to understand in how far observations obtained 601 under controlled lab conditions allow deriving conclusions 602 about the behaviour in real, highly fluctuating, outdoor 603 conditions. Therefore, its real usefulness will depend on 604 the creativity of its users.