Genetic network stability analysis

A living organism is an interacting system containing the genome and the expressome, defined as all the molecules (the transcriptome, proteome, metabolome) produced according to the genetic program and for which expression levels are regulated by genes and their epigenetic states in response to external influences or stresses (see Fig. 1). Likewise, the expression states of the genes are regulated by the components of the expressome. For the sake of model simplicity, but without loss of generality, we will specifically talk about the genes expressing as, and being regulated by, proteins. However, other levels of description including transcriptome, metabolome, etc. could also be viewed as relevant aspects of the expressome, similarly impacted by both endogenous and exogenous (environmental) factors. We start from the organism in a “normal” initial state, in which all genes have youthful/healthy expression profiles. With the passage of time, t, most of the genes retain “normal” expression profiles, while a few genes, e g (t) genes in total, subsequently become either damaged or (epigenetically) dysregulated and represent a few “defects” or “errors” in the genetic program. Gene transcripts are translated into the proteome, including its “defects”, at a certain translation rate p. Defects may appear in the expressome even if the genome state is perfectly regulated, due to unavoidable imperfections in translation or metabolic transformations17. Below, we assume that time-dependent quantities such as e g (t) may be averaged over times longer than the characteristic interaction times, but are still much shorter than the lifespan of the organism.

Figure 1: The minimum stability analysis model for a gene network. At any given time the genome consists of a number of normally expressed and dysregulated genes. The proteome accumulates “defects”, such as the proteins over- or under-expressed by dysregulated genes, which are removed via the protein quality-control or turnover systems. DNA repair machinery controls epigenetic states of the genes and restores normal expression levels. On top of this, interactions with the environment damage both the proteome and the genome subsystems, increasing the load on the protein-turnover and DNA-repair components. Parameters f, β, δ, p and c appear in Eqs. (1) and (2), and are interpreted in the text below. The figure was drawn by Peter Fedichev. Full size image

Next, we assume that the initial state of the genome is almost stable, meaning that the number of improperly expressed genes is small relative to the total genome size, and, therefore, the number of improperly produced proteins copies, e p (t), is also relatively small. This allows us to ignore the interaction between defects in the genome and the proteome. Accordingly, the most general model describing the dynamics of the interacting defects in the genome and in the proteome can be written in the following form:

Here , β is the coupling rate constant characterizing the regulation of gene expression by the proteins. The constant K is the average number of genes regulated by any single protein and represents a simple measure of the overall connectivity of the genetic network. The constant c reflects the combined efficiency of proteolysis and heat shock response systems, mediating degradation and refolding of misfolded proteins, respectively, whereas δ characterizes the DNA repair rate. The parameters—coefficients in Eqs. (1) and (2) are described by arrows on Fig. 1. Furthermore, the model includes the “force” terms, f p (t) and f g (t), which characterize the proteome and genome damage rates, respectively. The forces can represent any of a number of things, including oxidative stress (metabolic), temperature, gamma-radiation (environmental), that are imperfectly compensated by protective mechanisms.

Eqs. (1) and (2) can only hold in their simple linearized form if the total number of regulatory errors is small and the defects do not interfere with the repair machinery or any other rare essential subsystems of the gene network. If, for example, a defect alters a DNA repair system-associated gene expression or protein level, the repair rate drops, the system becomes unstable and may quickly diverge from its normal state18,19, as we show below. To avoid complications arising from introducing such nonlinearities, we adopt a simple hypothesis as to how defects in the evolving gene network could be responsible for the demise of a cell or organism. More specifically, we assume that mortality at any time is dependent on the probability of a defect to “land on” and damage or dysregulate an essential gene in a sufficient fraction of cells to lethally impair tissue function locally. Any gene in the model can be dysregulated for a short time (brief relative to lifespan) and then “repaired”, or its products acquired from healthy neighboring cells. Therefore, a gene is considered essential if the disruption of its expression, even for a limited time, is lethal to cells in which it was disrupted, once their occurrence exceeds some threshold fraction of cells. The suggested picture is quite general, however, and is easily extended to entire animals, since the stability boundaries are the same—only the exponents will be reduced because some threshold fraction of a tissue must die (in some most-vulnerable tissue type) to produce organism lethality. In this case the population dynamics of a set of gene networks representing N(t) organisms can be represented by:

where M(t) = ωe g /G is the mortality rate, proportional to the fraction of mis-regulated genes, e g /G, and ω is an empirical factor, roughly a measure of the (small) fraction of genes in the whole genome that are essential.

As shown in Methods, a general solution to Eqs. (1, 2) is a linear combination of two functions characterized by well-separated time scales. Most external perturbations lead to responses, which relax quickly at time scales on the order of the network’ s inverse relaxation rates, c and δ. However, life-long changes, such as aging or development, are usually considerably slower. Therefore we can use adiabatic approximation to obtain the effective equation for the age-dependent changes in the number or regulatory errors and, consequently, the mortality

Here F = f/(c + δ) is a combined measure of genotoxic stress, , and Λ = (βpGK − cδ)/(c + δ) is the exponent characterizing genetic-network stability, which is precisely the propagation rate of gene-expression-level perturbations. As we will see later, in the long run, stress levels can be averaged over longer periods and hence presumed to be time-independent, f p,g (t) = const. This yields the following expression for the age-dependent mortality rate:

The nature of the solution is very different depending on the sign of the exponent Λ. Whenever the combined efficiency of all repair systems is lower than a measure of the defect proliferation rate,

then the gene network is unstable, Λ > 0, and the number of regulatory errors (defects) in the genome and in the expressome grows exponentially along with mortality, M(t) ∼ exp(Λt), which is precisely the celebrated Gompertz law12. The Gompertz exponent, Λ, is related to the Mortality Rate Doubling Time (MRDT), t Λ = ln 2/Λ, whereas the average lifespan is given by

The quantity depends on both the exponent Λ and on the genotoxic stress level through the parameter γ = M 0 /Λ, where M 0 ∼ ωF/Λ is the mortality at birth or Initial Mortality Rate (IMR) in our model. For many species following the Gompertz mortality law γ is very small (γ ≈ 0.05 for fruit flies if computed using MRDT and IMR values from AnAge database11). Accordingly, the logarithm is usually large and the life expectancy, t le ≈ Λ−1 log(1/γ), greatly exceeds MRDT, the time scale characterizing the gene network instability. This is our precise definition of the Gompertz limit describing long lived species including humans, when the logarithm of a large argument is a very slow function, and the lifespan of the species is determined by the gene regulatory network properties only and depends very weakly on the genotoxic stress level through the value of γ. This also means that the lifespan does not depend on precise specification of stress levels or their variation over time. The same argument would hold if a small non-linearity is added in Eq.(4) and thus establishes a considerably wider applicability range for the basic linearized Eqs. (1, 2, 3).

A considerably more intriguing situation may occur when the genome is stable, Λ < 0 or R 0 < 1, and the gene network may remain stable under reasonable stress conditions for a very long time. The fractions of dysregulated genes and of misexpressed components of the expressome will then stabilize at constant levels, as will the mortality rate itself maintain the same level . Constant mortality rate means that the population of animals dies off exponentially rather than age-dependently: , which is much slower than the Gompertz-law prediction. We believe that the age-independent mortality observed in naked mole rat experiments over a very long lifespan7, together with exceptional stress resistance of naked mole rat tissues9, may be manifestations of this stable scenario. We predict that the gene networks of negligibly senescent animals are exceptionally robust, and the number of dysregulated genes will scarcely change with age. This argument is supported by the observations16 in which the number of genes differentially expressed with age was compared among naked mole rat, mice and humans.

Aging of the fruit-fly transcriptome

Analysis of model solutions (see Methods) suggests that aging in gene networks of “normally” aging or Gompertzian animals manifests itself as a highly correlated changes to the genome and the expressome states, occurring on distinct and well-separated time scales. We show that even though most external stresses lead to perturbation of the gene network, which relax quickly back to unperturbed levels, many experimentally measurable properties of the organism state should reflect the underlying instability, to an increasing degree as animals age. This means that gene-expression (or metabolite, etc.) levels should change with age in a coordinated manner and slowly deviate from their healthy/youthful states. We then asked whether our model is supported by gene-expression data from fruit flies (ref. 20). The measurements were performed at 6 different ages, for two groups of adult Drosophila melanogaster: normally (“ad lib”) fed control flies and calorically restricted (CR) flies. Figure 2 is a Principal Components (PC) analysis plot, in which each point represents the state of gene expression for one combination of age and diet.

Figure 2: Principal components analysis of gene expression profiles in aging flies (data from20), fed on control (ad lib) and Calorically Restricted (CR) diets. Every point represents a transcriptome for flies of a specific age and diet. As the animals age, the genetic network accumulates regulation errors and the transcription levels change in a single direction, up to a limit beyond which viability cannot be maintained. Full size image

Remarkably, aging in flies follows a unidirectional and thus apparently pre-defined or hard-wired in the genome trajectory of gene expression (along PC1) throughout their lifespans, accompanied by apparently rapid and random expression changes along orthogonal directions such as PC2. Variance along the PC2 axis is nevertheless small relative to the inter-group differences distinguishing ad lib from CR-fed flies. This may indicate that the corresponding transcriptional changes occur in response to, for example, nutrient-supply variation. Along the PC1 dimension, although there are stochastic contributions, there is a strong, systematic dependence on age in each of the two diet groups. Points on the extreme left correspond to the youngest flies, and points for older age-groups are displaced progressively to the right. Thus, deviation of the gene expression profile from the young state increases with age, indicating that the number and extent of dysregulated genes increases along with mortality up to a point when the accumulation of gene-expression abnormalities becomes incompatible with survival of the organism. This interpretation goes well in hand with the arguments used in derivation of Eqs. (1, 2, 3) and qualitatively support the presented model as a very general description of aging in gene networks of realistic animals, including multicellular organisms, such as fruit flies. We describe a generalization of the equations as well as the transcriptional and metabolic changes in aging flies along with their relation to the Gompertz mortality law in a subsequent work21. Both of the groups age in a similar way (along the same PC1 direction), but at a considerably different rate. We are leaving the detailed analysis of the aging trajectories differences between CR-fed and control flies for a future work.

Genetic-network stability and stress resistance

The proposed model may be considered as a general theory that subsumes previous “error catastrophe” theories19,18 as special cases. It was long considered that error catastrophes can be probed in experiment where the effects of various stresses on animal lifespan were observed22. To understand how the model presented above deals with stresses, we will first reanalyze a related experiment from23, in which flies of varying age were exposed to a traumatic brain injury (TBI) for a short time and then observed for a time T that is small compared to the lifespan of the animals. The mortality index MI T is calculated as the fraction of animals alive at the start of stress, dying over a short observation interval during or following stress application. The post-stress lifespan was also determined. To model the experimental settings we assume that the animals at an early age t 0 , , were subjected to an external genotoxic stress characterized by the amplitude Fs, which is proportional to number of traumatic strikes N in each experiment. As shown in Methods, a generic stress perturbs the state of the gene network and within a linear response theory, the mortality in the experiment has a contribution to both the slow and fast modes. At late ages, the influence of the stress mostly dies out due to the fast relaxation processes in the genome and the proteome. On the contrary, since the gene network of flies is unstable, the influence of a stress applied early in life is maintained in the slow mode and shortens the lifespan of the animals. The mortality M as a function of time t and of number of traumatic events, such as strikes, N is given by

where is an empirical factor, proportional to the number of the strikes, , where χ is a stress and a species-dependent constant. Accordingly, we predict that the difference ΔM 0 between mortality in flies, exposed to a different number N of traumatic strikes, and mortality of the flies in the control group is proportional to N at any given age. Figure 3a shows a direct comparison of the mortalities of the treated and the control groups obtained from our analysis of the data from23. An alternative way to see this is to calculate the model prediction for the average lifespan as a function of N

Figure 3 (a) ΔM 0 , the mortality change as wild-type flies are exposed to N traumatic insults, is plotted as a function of N. (b) Average lifespan, t avg , is plotted as a function of N. In both panels (a,b), the solid lines indicate the theoretical prediction based on Eq. (6) and parameters estimated from experimental data. Grey symbols are experimental mortality data points. Full size image

and compare it with the experimental lifespan in groups with various stress levels. The results of the analysis of the population dynamics data from23 are presented in Fig. 3a,b, and show a fair agreement between the dependences observed in the experiment and the simple model predictions.

Even though the mortality at late ages does not depend on the fast gene network dynamics, the Mortality Index itself contains both kinds of contributions,

where E = (c + δ) is the effective relaxation time in the expressome (see the analysis leading up to Eqs. (15) and (19) in Methods). Empirically, in TBI experiment and therefore very strong stressors are required to produce a measurable change in the lifespan. This may be extrapolated to and compared with findings reported earlier22, where Drosophila adults in experimental groups were treated for 3–5 days with a number of agents shown to increase misincorporation into protein or RNA, at doses leading to <20% mortality. Although these treatments produced error rates much higher than were seen in the course of aging in control flies, and produced a small mortality increase during the treatment, the misincorporation rates subsequently returned to control (pre-treatment) levels. This would be consistent with predictions of our model, if most of the stress-induced perturbations relaxed quickly to a level nearly indistinguishable from controls. Because the overall Mortality Index in the experiment was small, the average lifespans of survivors were indistinguishable from controls22. The effects of the stress on the lifespan may have been further reduced by hormesis, a known ability of weak stresses to improve survival of animals24,25,26. The latter is clearly a non-linear phenomenon, which is not very strong in the Gompertz limit in any case and can not be explained by the suggested simple linearized model.

Extreme longevity has long been associated with exceptional resistance to a variety of stresses27. And conversely, the decrease of stress resistance with age is one of the best-established indices of aging. The relation should be taken with caution, since the stress resistance measured by the Mortality Index and described by Eq. (10) contains the contribution of processes occurring at all time scales. Only a measure of stress resistance associated with the response of the slowest modes of the gene network can be related with aging and longevity. A curious situation may occur when Λ < 0 in Eq. (6), indicating that the efficacy of repair systems is high enough to prevent exponential system deterioration with age. This would also imply robust resistance to stresses, consistent with the exceptional stress resistance of negligibly senescent species9. For example, a comparison of survival between negligibly senescent vs. short-lived clam tissues treated with tetr-butyl hydroperoxide showed much higher resistance to oxidative stress in long-lived clams3. Also Oxygen Radical Absorbance Capacity (ORAC) was measured in young and old clams of both types. The experiment showed an age-related decline in ORAC for shorter-lived clams, whereas ORAC did not change with age in tissues of negligibly-senescent clams, which is entirely in line with the model predictions, since there are no possible changes in the gene network state leading to a deterioration of the genotoxic compensation abilities, if the gene network of an organism operates in the stable zone.