Periodic ethanol supply as a path toward unlimited lifespan of Caenorhabditis elegans dauer larvae

The dauer larva is a specialized stage of worm development optimized for survival under harsh conditions that have been used as a model for stress resistance, metabolic adaptations, and longevity. Recent findings suggest that the dauer larva of Caenorhabditis 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 a 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 suggested 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 feeding frequency and amplitude affect lifespan extension. Based on the comparison of the model with experimental data for fixed ethanol concentration, we proposed the range of feeding protocols that could lead to even longer dauer survival and it can be tested experimentally.

Initial conditions are motivated by experimental conditions.It is natural to consider the mitochondria activity to be unity since there is no damage at the beginning.Following similar reason the initial toxic level is set to zero.According to the simulations, the acetate level is a fast variable which converge to a certain balance value irrespective of the initial condition, and thus its initial value is not much relevant as long as it starts above the starvation limit.The initial lipid storage is determined similar to parameters in the model, where the value leading to the best fitting is taken.
The curves in Fig 8, are calculated by parameter scanning, were for each given pair of (ω E , A) we run the simulation with a set of initial lipid storages equally spacing from low to high, and the "range" w corresponding to this pair of (ω E , A) is given by the maximal value of w from the simulation result of this set of initial lipids.Scanning over initial lipid storage rules out the effect of initial conditions, as we want to explore the ability to have an unlimited lifespan irrespective of the initial conditions.
In total, the model contains 10 parameters and needs 4 initial values for the variables.To fix the parameter values we need to reproduce survival curves of 3 strains for 2 ethanol conditions (where only selected parameters are altered to recapitulate the mutant situation).In approximating the lifespan curves, we aimed to satisfy both the position of the characteristic lifetime (when survival drops to 50%) and the steepness of the survival curve decay.Additionally, we have certain biologically motivated limitations on the initial values and rates.While we have not performed a systematic scanning of the parameter space to identify suitable parameter sets, we suggest that the number of imposed constraints allowed us to converge to a unique set.Furthermore, for the case of periodic ethanol supply, we performed a proper sampling over the range of possible initial conditions to rule out the effect of the starting values.

DIVERGENCE OF THE LIFESPAN AT ONE SIDE
In Fig 7, there is a region of j 0 values considered to lead to an unlimited lifespan.When we slowly increase the value of j 0 across the lower boundary or decrease the value of j 0 across the higher boundary, the value of lifespan experiences a switch process from limited value to an unlimited one.This transition from limited to unlimited lifespan can be of two types: continuous divergence or a discontinuous jump.Here we briefly discuss which type the switch occurs for both the lower and higher boundaries.We first focus on the high ethanol boundary which we denote j re .
Figure S1.When j 0 < j re , the toxic level grows and stays below its threshold value c ≲ c h and the dauer survives (a).When j 0 > j re the toxic level grows higher than the threshold such that the dauer is damaged and dies (b).
The dynamics around but below the transition suggests that the trajectory c of the toxic compound c approaches a certain baseline value css close to c h exponentially.
This approximation is supported by numerical simulations: the value of log(c ss − c)/c h is shown in Figure 12.Taking the logarithm of equation 9 yields a logarithmic divergence of time t for toxic compound to go across the threshold c h .With considering all relevant parameters the expression of lifespan t ls is thus: where j re is the transition point and κ is constants while t damage is the time dauer could survive when irreversible damage happens.Numerical simulation of lifespan t ls as function of the distance to boundary j 0 − j re is shown in Fig 12, which suggests that logarithmic function fits the functional form of divergence quite well.The switch process at the left edge j le for small ethanol influx is challenging to treat analytically, but the numerical simulation up to precision limit indicate the switch process at j le to be a discontinuous jump, see  Frontiers

ANALYTICAL EXPLANATION OF THE DEPENDENCE OF RANGE ON PARAMETERS
Here we explain why w(ω E , A) takes the functional form shown in Fig 8 through an analytical study.As discussed before, detoxification of the toxic compound at low acetate level and mitochondria regeneration at high acetate level must happen in alternating manner to allow for an unlimited lifespan.
Mathematically, in the state of the unlimited lifespan, the time dependent acetate level a(t) should satisfy following relations, which corresponds to the level of acetate generating minimum "energy" flux j in .
where the maximal and minimal are taken at large t when the initial condition is fully forgotten.In other words, the threshold value a min = j m /k 4 must be contained in the interval (min[a], max[a]), as shown in Fig 7 .The range w is defined as the size of the interval where we can alter j 0 while keeping the lifespan of dauer to be unlimited.If we can estimate the dependence of max[a], min[a] on j 0 , the above equations can be used to evaluate the range w.Here we assume that altering j 0 by a small constant δj will only change the value of a(t) and thus its maximum and minimum in a linear way: where α is the proportionality constant assumed to be the same for both min[a] and max [a].If we start with a j 0 such that equations S3 and S4 are satisfied, the value of δj we can choose such that equation S3 and S4 are still satisfied is given by: An estimated relation between "range" w and the acetate fluctuation region size (max[a] − min[a]) is thus given by: where A acetate defined in above equation is considered as the fluctuation amplitude of a(t).Of course, that the profile of a(t) does not vary with j 0 such that min[a] and max[a] share the same α is only a rough approximation.Numerical simulation estimating w and A acetate for different feeding frequency ω E suggests that equation S10 holds well when A acetate is small, as shown in Fig S5.A linear analysis at high feeding frequency would help to further explain the relation between A acetate and feeding parameters A and ω Et .
Under sinusoidal feeding protocol with high frequency, it is observed that also a, l and c behave almost sinusoidally and oscillate around their average value: The non-constant parameter k 1 in (6) and k 2 in (7) can be expanded as linear functions of l:

Figure S2 .
Figure S2.Around j re , the trajectory of chemical compound c approaches the quasi-steady state c ss ≤ c h exponentially.c h is the threshold value of toxic compound above which mitochondria start to damage.

Figure S3 .
Figure S3.Logarithm fitting (red dashed line) of the divergence of lifespan at high ethanol edge.The factor 1 κ fits to 11.5/k 4 .

Figure S4 .
Figure S4.Jump of lifespan at low ethanol edge.

Figure S5 .
Figure S5.Width w as a function of acetate amplitude A acetate for different feeding frequency ω E with fixed A.