Periodic ethanol supply as a path towards unlimited lifespan of C.elegans dauer larvae

The dauer larva is a specialized stage of development optimized for survival under harsh conditions that has been used as a model for stress resistance, metabolic adaptations, and longevity. Recent findings suggest that the dauer larva of C.elegans may utilize external ethanol as an energy source to extend their lifespan. It was shown that while ethanol may serve as an effectively infinite source of energy, some toxic compounds accumulating as byproducts of its metabolism may lead to the damage of mitochondria and thus limit the lifespan of larvae. A minimal mathematical model was proposed to explain the connection between the lifespan of dauer larva and its ethanol metabolism. To explore theoretically if it is possible to extend even further the lifespan of dauer larvae, we incorporated two natural mechanisms describing the recovery of damaged mitochondria and elimination of toxic compounds, which were previously omitted in the model. Numerical simulations of the revised model suggest that while the ethanol concentration is constant, the lifespan still stays limited. However, if ethanol is supplied periodically, with a suitable frequency and amplitude, the dauer could survive as long as we observe the system. Analytical methods further help to explain how the feeding frequency and amplitude affect the lifespan extension. Based on comparison of the model with experimental data for fixed ethanol concentration, we propose the range of feeding protocols that could lead to even longer dauer survival and can be tested experimentally.

across its body boundary shuts down. As a result, it was long believed that C.elegans 10 dauer survive solely on stored lipids and are not able to uptake any carbon source form 11 their environment [3][4][5]. However, our recent findings [6] showed that C.elegans dauer 12 can utilize ethanol as an external carbon source, see Fig 1. Remarkably, at optimal 13 concentrations, ethanol could expand lifespan of dauer larvae two fold for a wild type 14 and up to four fold for some mutants. Ethanol can penetrate across the cuticle, and 15 thus gets channelled in the metabolic pathways of C.elegans dauer larvae. The enzymes 16 responsible for the first metabolic steps are SODH-1 and ALH-1, that transform ethanol 17 to acetate which can be activated into acetyl-COA and enters the major metabolic 18 pathways of TCA cycle, glyoxylate shunt gluconeogenesis and and lipid metabolism, 19 thus augmenting the metabolic pathways that dauers use for energy production [5][6][7][8][9]. 20 SODH-1 and ALH-1 are found to be up-regulated in the presence of ethanol, whereas in 21 sodh-1 mutant, the ethanol is no longer incorporated and does not affect the lifespan of 22 dauer. Experiments with radioactively-labelled ethanol have shown that it can be 23 utilized for the production and accumulation of stored lipids, thus providing an 24 effectively unlimited source of energy to dauer larvae in case of permanent ethanol 25 supply [6]. This led us to the question of why even in the presence of this energy source dauers 27 do exhibit a longer lifespan but then eventually die. To help to answer this question we 28 proposed a mathematical model describing the relation between the lifespan of C.eleans 29 dauer and the supplied ethanol, based on the known metabolic pathways of dauer larvae. 30 We assumed that the dauer dies either due to the lack of energy or due to the 31 accumulation of some, not yet identified toxic compound(s) [6] that could resemble the 32 so-called "lipitoxicity" factors in mammalian systems [10][11][12]. As experimentally 33 observed, the death of worms was preceded by deterioration of mitochondria. We also 34 assumed that these two mechanisms lead to mitochondria damage and then to death.

35
This model was very successful in explaining experimental data on lifespans of dauer 36 and various mutants in presence and without ethanol.

37
While identifying the exact toxic component that limits the lifespan of dauer is still 38 an ongoing research project, we were interested to explore whether or not the lifespan 39 could be extended even further. To this end, we assume there are two self-recovery 40 mechanisms, namely regeneration of mitochondria and detoxification, and test what 41 they lead to. These two mechanisms alone still result in dauer's death if feeding 42 protocol is constant. However, when we use periodic supply of ethanol in the model, an 43 unlimited lifespan can emerge according to the numerical simulation. By comparing 44 model predictions with existing data, we also suggest feeding protocols that now can be 45 directly tested in future experiments on dauer.
Mathematical model 47 Fig 2. Schematics of the mathematical model of the metabolic network of C.elegans dauer larvae. Externally supplied ethanol is transformed into the acetate with the rate j in , which can be either constant or varies in time depending on the supplied ethanol concentration. Acetate (ch. acetyl-CoA in Fig 1) is then used either for energy production and carbohydrate synthesis or can be stored in lipids. Mitochondria takes damage from lack of carbohydrate production or accumulation of toxic compounds as the product of lipolysis.
A simple model of the metabolic network of C.elegans dauer larvae was introduced in [6] 48 and accurately reproduced the lifespans of dauer with and without ethanol for wild type 49 worms as well as for various mutations. The framework of the model follows the largely 50 coarse-grained metabolic pathway of dauer. All the chemical components falling into 51 the category of "available energy" are combined and called "acetate", which is the 52 central representative component of this category. Similarly, the components 53 corresponding to "stored energy" and "consumed energy" are denoted as lipids and 54 carbohydrates respectively. Acetate and lipids could transform between each other as 55 the balance between free and stored energy. At the same time, acetate continuously 56 transforms into carbohydrate unidirectionally to support the main functions of an 57 organism including mitochondria. If production of carbohydrates drops below a certain 58 minimal threshold, mitochondria start to get damaged and the dauer dies. In presence 59 of ethanol, acetate gains an influx proportional to its concentration. During the process 60 of releasing stored lipid, toxic compounds are produced as a side product, as the second 61 major reason damaging the mitochondria alongside the lack of carbohydrate production. 62 Our model also included the effect of genes that were identified as regulatory factors 63 through genetic experiments with loss-of-or reduction-of-function mutations [6]. conditions. We will refer to it as a control and use it as a proxy for the wild-type dauer 66 under stress, since the experimental results show very little difference between daf-2 67 and wild-type dauers. Loss-of-function mutation in the aak-2 /AMPKα in 68 daf-2(e1370);aak-2(gt33) double mutants causes an enhanced lypolisis rate, which leads 69 to a reduced lifespan as compared to control strain under both feeding conditions with 70 and without ethanol [7,13]. A reduction-of-function allele in the class I PI3-kinase 71 age-1(hx546), on the other hand, is supposed to have the reduced lipid synthesis rate.

72
This assumption is based on experimental results with the mutant age-1 in dauer 73 state [6]. Its lifespan is similar to control dauer when there is no ethanol supply, but has 74 a large increase when the ethanol is supplied [6].

75
The goal of this work is to identify potential ways of how the dauer could survive 76 even for a longer time. Thus here we consider mechanisms by including which the model 77 will be able to produce an unlimited lifespan while still remaining consistent with the 78 results of the previous experiments. There are two essential and rather natural 79 mechanisms that have been omitted in the original model [6] while having a potential 80 for lifespan extension: detoxification [15,16] and a possibility for mitochondria to 81 regenerate [17][18][19] (see green arrows in Fig.2). We will show in the following, that the  To demostrate this, we first formalize the schematics in Fig.2 into the system of 85 ordinary differential equations that describe the chemical reaction network of ethanol 86 metabolism [14]: Here "a" and "l" denote the concentrations of the acetate and lipids respectively, production k 4 a falls below the minimal required "energy" flux j m , or with a rate k d2 100 when the toxic compound c accumulates above a certain threshold concentration c h (Θ 101 in the equation is the Heaviside step function). There are many known mechanisms of 102 mitochondria surveillance and maintenance [17][18][19]. Here for simplicity, we suggest a 103 phenomenological law of mitochondria recovery, where the mitochondria regenerates its 104 current damage level (1 − m), provided it is not suffering from any further damage with 105 a constant regeneration rate k r .

106
While most of reaction rates in the above equations are considered constant for 107 simplicity, some rates do depend on variables. First example is the linear dependence of 108 k 4 on m, which assumes that the energy production requires functional mitochondria: wherek 4 is a constant. Another non-constant rate is k 1 quantifying acetate-to-lipids 110 conversion. It reflects the fact that each dauer has a storage limit capacity l s (it cannot 111 accumulate unlimited amounts of lipids): Here l 1 is a characteristic lipid concentration at which the conversion starts to saturate 113 andk 1 is a constant. Finally, we also assume that k 2 has the functional form of Michaelis-Menten reaction [14] 115 wherek 2 and l 2 are constants. If in the above equations, we set k r = 0 and k c = 0 we 116 will recover the system studied in [6].

117
November 12, 2021 4/20 The model including self-recovery mechanism, however, should also reproduce 118 lifespan of dauers with and without ethanol, as well as different mutants as was observed 119 experimentally [6]. This also means that this model should result in finite lifespan under 120 constant ethanol supply. However, the novel possibility for lifespan extension may now 121 emerge for a non-constant feeding, where the supplied ethanol concentration varies in 122 time, for instance, according to a sinusoidal protocol. We next show by using numerical 123 simulations that the model reproduces experimental observations under constant feeding 124 and predicts the lifespan extension under periodic feeding protocol. chosen by checking whether the lifespan ratios between mutants (daf-2, daf-2;aak-2 and 130 age-1) with and without ethanol generated by simulations fit the previous experimental 131 results [6]. When one set of parameter is considered as the control strain without The detailed dynamics of one of the parameter sets, for control strain with and 142 without feeding is shown as an example in Fig 4. When there is no feeding, dauer brakes 143 down storage lipids to keep its acetate level and thus the carbohydrate production rate. 144 As the lipids run out, the mitochondria is damaged for lack of carbohydrate production 145 and results in the death of dauer. When ethanol is supplied at a sufficient level, 146 starvation becomes no longer a concern. However, the toxic compound continuously 147 accumulates and as it goes beyond the threshold at some point, the mitochondria start 148 to take damage and finally the larvae die. The details of the simulation including the 149 numerical methods [14,20,21] and parameters are provided in the Appendix I. . Without ethanol supply the dauer consumes the stored lipids and dies due to starvation, while with ethanol the dauer dies due to accumulation of toxic compounds.
As the above two examples show, starvation or accumulation of toxic compounds is 151 the reason for mitochondria damage and the resulting death of larvae. We can 152 demonstrate more generally the condition for the finite lifespan of dauers for a constant 153 ethanol concentration.  infinite lifespan when the ethanol concentration is sufficiently high or low, respectively. 157 If, however, these two lifespan vs. influx curves intersect at some value of j in , lifespan 158 will always remain finite, as for any given ethanol concentration and the corresponding 159 influx, there will be at least one reason that the dauer dies within limited time 160 determined by k d1 or k d2 .

161
If the two curves (with each of the damage removed) do not intersect, and they thus 162 would form boundaries of a domain in between where the value of j in would support an 163 infinite lifespan. As we mentioned above, in experiments, the dauer survives always the 164 finite time in presence of ethanol, thus defining for us the parameter range that has to 165 be chosen in simulations. 166 2 Periodic ethanol supply 167 The above results show that ethanol supply keeps mitochondria operational, but the 168 accumulating toxic compounds damage the mitochondria. Here we hypothesize that 169 periodic ethanol supply might be the key to an unlimited lifespan of dauer. While 170 periods of supply might be used to replenish lipid storage and repair mitochondria, the 171 periods of no feeding can be used to degrade the accumulated toxic compounds. We 172 now test this hypothesis numerically. For simplicity, we use a sinusoidal feeding protocol 173 with a feeding amplitude A, feeding frequency ω E and a positive baseline value j 0 : With a proper parameter choice the numerical simulations of the model show that the 175 mitochondria damage and regenerate periodically until the end of simulation, no matter 176 how long these last.
177 Fig 6. Example of a lifespan extension as a result of periodic feeding. The baseline value j 0 /j m = 0.8 , while A = j 0 . The feeding frequency ω E is the same as the oscillatory frequency of acetate level a shown in blue. The inset shows oscillations of toxicity near but below the toxic limit, while mitochondria are almost normal.
Indeed, this situation becomes possible when parameters are tuned such that the 178 periodic feeding permits the worm to accumulate toxic compounds while intaking 179 ethanol and fuelling mitochondria but then remove them with a diet at the cost of some 180 mitochondria damage, which can, however, be regenerated during the next intake cycle. 181 These simulations suggest that periodic feeding protocol does provide a theoretical 182 possibility of an unlimited lifespan extension (Fig 6). We next investigate in more detail 183 how this effect depends on model parameters.  According to the simulation results, we see that the lifespan extension is possible 209 when the feeding frequency is within a certain interval. The experiment with the 210 feeding frequency ω E corresponding to the maximal range w is expected to give highest 211 chance to observe the effect. The simulations also suggest that the range w grows with 212 feeding amplitude A in an almost linear way if the feeding frequency is high enough.

213
This is because at high frequency region the range w is proportional to the oscillation 214 amplitude of acetate, which can be explained by an approximate analytical solution.

215
(For details of the analysis, see Appendix. II.)

216
The range-frequency curves can potentially help us to identify suitable feeding 217 frequency and amplitudes for which it is most likely to observe lifespan extension in 218 experiments. To do so, we still need to connect our mostly dimensionless equations to 219 realistic parameters. This is not too straight-forward since not all parameters of the 220 enzymatic kinetics as well as chemical concentrations in the dauer were measured yet. 221 However, for the case of the feeding frequency, we may take a short-cut, where we can 222 determine the timescale by equalling the control lifespan without ethanol in the model 223 defined as time where m falls to, for example 0.5 to that in experiment defined as 224 respective 50% survival and restore all reaction rates in real time units. Also the feeding 225 amplitude is simply set as large as possible (see below) so there is no more information 226 needed. Fig 9 shows the range vs period (given in hours) relation for control strain and 227 daf-2;aak-2 mutants under maximal feeding amplitude. The maximal feeding amplitude 228 is defined as A = min[j 0 ] (i.e. the smallest j 0 among all j 0 used in scanning, such that 229 the influx j in is always positive). Another definition A = j 0 for all j 0 is also possible 230 and leads to similar results. 231 Fig 9. Range of the baseline ethanol supply levels where we expect to see unlimited survival of dauer larvae shown for control and daf-2;aak-2 strains as a function of the feeding period T = 2π/ω E . Here it is assumed that the amplitude A takes the value of min[j 0 ].
These simulations not only suggests an optimal feeding period for both strains, but 232 also indicate that daf-2;aak-2 mutant is a better option for experiment, for not only the 233 larger range value but also the smaller optimal feeding period. It requires feeding period 234 of the order of 10 hours (so the media for larvae can be changed once a day) and the 235 effect should be seen much earlier, as the original lifespan of daf-2;aak-2 is much shorter 236 and thus overall a shorter experiment could be carried out.

238
Previously we have shown that the lifespan of C. elegans dauer larvae can be greatly far, however, we have neglected the possibility of mechanisms that help dauer to recover 243 from this damage. Therefore, two biological self-recovery mechanisms, namely the 244 detoxification and mitochondria regeneration, were introduced into the model. 245 Importantly, despite self-recovery mechanism, for constant ethanol supply, model 246 reproduces the experimental observations of extended but limited lifespan.

247
However, when feeding protocol is periodic, an unlimited lifespan can emerge. The 248 possibility of the unlimited lifespan can be explained by the switch between two feeding 249 phases, where the first one at high ethanol concentration repairs the mitochondria at 250 the cost of toxic compounds accumulation while the second one, at low ethanol 251 concentration, has the toxic compounds degraded but also damages the mitochondria 252 slightly. For this process to keep the dauer surviving thus requires both, mitochondria 253 regeneration and toxic compounds detoxification mechanisms, to function.

254
To characterize the unlimited lifespan predicted by the model systematically, we 255 defined a range of baseline feeding fluxes, which quantifies the ability of a certain set of 256 feeding parameters to support the unlimited lifespan. The dependence of this range on 257 feeding frequency and amplitude were studied numerically with some supporting 258 analytical arguments. This dependence combined with previous data helped us to 259 suggest suitable feeding period and amplitude that can now be tested experimentally. If 260 lifespan extension of dauer larvae appears under periodically supplied ethanol, that 261 could be a confirmation of our hypothesis about the existence of recovery mechanisms. 262 This study treats the identity of the toxic compounds open and does not specify the 263 concrete mechanisms of mitochondria recovery and detoxification. Ultimately for our 264 comprehensive understanding of dauer larvae lifespan extensions mechanisms and 265 generalization of those to other organisms we need to push towards identifying the exact 266 biological players of toxicity and recovery competition.