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On the one hand, adaptation beyond genetic changes—for example epigenetic adaptation—is a well-documented phenomenon in the bacterial world [ ] and is the focus of intense research activity [ 27 , — ]. In particular, such phenotypic variability can provide a faster and more flexible type of response than the one associated with traditional genetic mutations.

Nevertheless, it is important to underline that Fridman et al. In particular, they found mutations in genes controlling the so-called toxin-antitoxin circuit, mediating the response to antibiotic stress [ 24 ]. Thus, strictly speaking, our modeling approach constitutes an effective or phenomenological approximation to the more complex biology of this problem. This observation opens promising and exciting avenues for future research to shed light on how broad probability distributions of lag times—possibly with heavy tails—can be actually encoded in phenotypic or genetic models.

Actually, scale-free power-law distributions of bacterial lag times have been recently reported in a specifically-devised experimental setup [ ]. Similarly to our conclusions, this work also emphasizes that a broad distribution of individual-cell waking-up rates is needed to generate non-exponential decays of the overall lag-time distribution.

Similarly, another exciting possibility would be to develop computational models akin to the phenotypic one proposed here but implementing genetic circuitry; i. Future developments and perspectives. In future research, we would like to further delve onto several aspects, both biological and theoretical, of the present work. As a first step, we leave for forthcoming work the analysis of the pertinent question of how similar systems respond to randomly fluctuating environments as opposed to periodically changing ones; do they develop heavier tails to cope with such unpredictable conditions in a sort of bet-hedging strategy?

How do the statistical features of the environmental variability translate into the emerging lag-time distributions? From a more theoretical perspective, we leave for an impending work the formulation of an extension of our approach that fully accounts for finite-size effects, thus introducing the next-to-leading order corrections to the generalized-Crow-Kimura macroscopic equation accounting for demographic fluctuations. Within this context, treating the variation-amplitude itself as an evolving trait is also a potentially fruitful route for further studies.

Finally, as a long-term project we plan to develop models and analytical approaches, similar to the ones explored here, but focusing on genetic evolution, employing explicit genotypic-phenotypic mappings, rather than just on phenotypic changes. In particular, by introducing this further layer of complexity it would be possible to generate more general types of single-cell lag-time distributions, not limited to exponential ones as the purely Markovian approach considered here.

Let us recall that a more general stochastic non-Markovian framework—i. Methods Numerical values of the parameters In order to fix parameter values we employed the experimental values and measurements in [ 24 ] as closely as possible. Initially the number of cells in the growing state is fixed to be equal to the carrying capacity K; thus no cell is initially in the dormant state.

The death rate for natural causes i. Variation functions We consider two different variation kernels for lag-time variations : the additive one,. In particular, the additive case reads: with where Erfc stands for the complementary error function , while in the multiplicative case, we consider with.

To obtain results for the transient state we determined the histogram after running for 10 cycles. On the other hand, to determine the steady state, we started measuring after cycles to make sure that a steady state has been reached and then collect statistics up to cycle , at intervals of 10 cycles to avoid correlations. We repeated the process for 30 realizations and calculated the histogram as well as the associated cumulants. Numerical integration of the macroscopic equation The parameter set and initial condition for numerical integration of the mean-field equations are the same as specified above.

Numerical integration was carried out using the finite differences method. Note that, when we calculate the probability distributions during the simulation, we must use the same bin size to be able to correctly compare with the theoretical distributions later. Supporting information S1 Text.

S3: Small Variation Approximation. S5: Spontaneous shifting to the dormant state. S6: Additional Figures. S7: Movies. PDF S1 Fig. Validity of the small variation approximation in the additive case. The first variation value is the best fit the experimental results main text , while the second causes negligible border effects. In both cases the small-variation approximation is a good approximation. EPS S2 Fig. Range of validity of the small variation approximation in the additive scenario.

Systematic comparison of Eq. S and Eq. This is not the Black Scholes model —VIX is all about "implied" volatility and measures the market's expectations for volatility over the coming 30 days. Because there is an insurance premium in longer-dated contracts, the VXX experiences a negative roll yield basically, that means long-term holders will see a penalty to returns.

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The SVXY has a slightly higher expense ratio of. A critical key for investing in SVXY is understanding that the fund is only intended for short-term trading and is not a buy-and-hold strategy. SVXY seeks its inverse return from its underlying benchmark for a single day, as measured from one net asset value NAV calculation to the next. Investors in SVXY should monitor and manage their investments daily. Inverse ETFs held for more than a day can lead to significant losses. These funds can be expected to perform very differently from the VIX.

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#### The study protocol was designed in accordance with guidelines of the Declaration of Helsinki.

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