Epigenetic Ratchet: Spontaneous Adaptation via Stochastic Gene Expression

Adaptation to unforeseen environmental changes is one of the most prominent features that characterize the living system. Although signal transduction and gene regulation networks evolved to adapt specific environmental conditions that they frequently experience, it is also reported that bacteria can modify their gene expression patterns to survive a huge variety of environmental conditions even without such pre-designed networks to adapt specically to each environment. Here we propose a general mechanism of cells for such “spontaneous” adaptation, on the basis of stochastic gene expression and epigenetic modication. First, a variety of gene expression states that are marginally stable states are generated by epigenetic modication. Then by taking advantage of stochastic gene expression and dilution by cellular growth, it is shown that, a gene expression pattern that achieves greater cell growth is generically selected, as conrmed by simulations and analysis of several models. The mechanism does not require any design of gene regulation networks. General relevance of the mechanism to cell biology is also discussed.

rate depends on the expression level, and cellular states with different growth rates are generated depending on modification levels.
In the present paper, we demonstrate that adaptation to achieve faster growth for given environmental conditions is generally attained only by noise and the growth-induced dilution, without preparing specific gene expression networks with multiple attractors. We will provide a general concept of selection from and discuss their relevance to cell biology.

Model
Here, we study a simple cell model consisting of proteins and the degree of epigenetic modification levels of genes. In the model, we take into account the following processes: (i) Each protein is synthesized at a certain rate according to the expression level of genes; (ii) proteins are spontaneously degraded; (iii) Some specific proteins determine the growth rate of the cell, whereas proteins are diluted due to the volume growth of the cell (the cell division is incorporated in the dilution process). (iv) For specific genes, epigenetic modifications are taken into account and can alter the expression levels of genes, whereas the modification changes need specific molecular machinery. Now, we introduce a simple model consisting of the expression level of two types of genes and the corresponding proteins: target (t) and sensor (s) genes and proteins. Here, each expression variable does not necessarily represent a single gene or protein but can be regarded as a collective variable representing the average behavior of genes or proteins that share similar functions.
The target protein contributes to the growth of the cell, whose rate μ, thus, μ is the function of its concentration. As the simplest example, we suppose that the synthesis rate of the target protein is determined by an epigenetic modification level, while the sensor protein is produced in a constant rate, v s .
Although the molecular mechanisms that regulate the epigenetic modification have not yet been fully elucidated, it is natural to assume that the epigenetic modification is regulated some other genes, which we call here as the sensor gene. If the concentration of the sensor protein exceeds a threshold value, it facilitates the synthesis of some proteins that modify the epigenetic state of the target gene. By means of the rate equations, the dynamics of the above model are given as s s s s s with where E t , P t , and P s represents the level of the epigenetic modification, the concentration of the target protein, and the concentration of the sensor protein, respectively. P t is produced in proportional to the epigenetic modification level, with a rate v t E t (the proportionality is not essential to the present study, as long as the production rate depends on E t . For example, the mechanism works when v t E t is replaced by v t E t /(1 + E t ) or v t /(1 + E t ), as shown in Supplement). d s and d t represent the degradation rate of the corresponding protein. l + and l − is the activation and inhibition rate of the epigenetic state E t , respectively. The modification by the sensor protein is represented by H(P s ) given by a sigmoidal function, , where θ is the threshold for the induction, and β is steepness of this increase, corresponding to the Hill coefficient (If H is replaced by a Hill function, the results are not altered qualitatively, as shown in Supplement). This form is adopted as a natural consequence of binding-unbinding kinetics between a DNA molecule and transcription factors, with the inclusion of the effect of cooperativity.
The target protein is synthesized at the rate proportional to the epigenetic state of the gene. Here we set specific growth rate μ to μ μ = + − ⁎ P P /(1 ( ) ) tain the cell growth). The results shown below, however, do not change qualitatively if we choose other forms of μ (see Supplement). Note that any specific input to increase the growth rate is not introduced here. In the Eqs. (2) and (3), a gene and corresponding protein to modify the epigenetic state are not explicitly included to the model for simplicity's sake. We have also studied a model in which the modifier gene and protein are implemented, and have confirmed that the simplification is not necessary to obtain the main results.
To study the influence of stochasticity on chemical reactions, we employed stochastic simulations of the reactions corresponding to Eqs. (1)(2)(3), via the Gillespie algorithm 14 (i.e., in the continuous limit with a large number of molecules Eqs. (1-3) is regained). The master equation which we used for the simulation is shown in the Supplement.

Results
Spontaneous adaptation via stochastic gene expression. In Fig. 1(a), examples of the time course of the growth rates and the variables are plotted for two different values of threshold θ, where the initial value of E t and P t at the steady-state values of Eqs. (1-3) with θ = 0, so that the growth rate is low initially. The time courses show that the growth rate of a cell switches to a high value when θ is large, whereas for small θ (black curve), it remains at the initial low value. In the former case, the epigenetic state of the target gene is modified so that the cell achieves a high growth rate. Dependence of the averaged growth rate on θ is plotted in Fig. 1(b), which shows that the growth rate starts to rise at approximately θ ≈ 150 and reaches the maximal value at θ ≈ 200. Note that even if we start from a fast-growth state, the cell cannot maintain the fast growth when θ ≈ 0. Thus, the failure of this adaptation at θ ≈ 0 is not due to the choice of the initial state.
When we set θ and the noise level appropriately, the adaptation mechanism works. It is intuitively explained as follows (referring the right panel of Fig. 1(a) might be useful): If the growth rate of the cell is low (i.e., P t is far from * P t ), then the sensor protein accumulates and P s exceeds the threshold value for H(P s ), θ, and then the epigenetic state of the target gene is modified. If the epigenetic state of the target gene reaches an appropriate value to allow for the increase in the growth rate, then all proteins are strongly diluted accordingly. Then, the concentration of the sensor protein hardly exceeds the threshold value because of the strong dilution by rapid growth, so that the modification of the epigenetic variable takes place at a low probability. As shown in Fig. 1(c), the probability of the change in the epigenetic variable suddenly drops to almost zero when μ is increased. Thus, the epigenetic state of the target gene remains at a certain value which supports high-growth rate. If the cell remains at a slow-growth state, however, then the epigenetic modification state keeps on being changed by noise. Hence, the system sooner or later reaches a fast-growth state and stays there.
Three conditions are essential for this adaptation mechanism to the fast-growth state work. (i). the noise level, due to the finiteness of the molecule number in a cell. With the decrease in the number, the noise level due to a stochastic reaction process is increased. In Fig. 1(d), the average growth rate is plotted against the average number of molecules. If the number is too small, noise dominates over the average gene expression dynamics, so that the cellular state fluctuates almost randomly. On the other hand, when the number is too large, the noise level is so small that the state is hardly kicked out from the initial low-growth attractor of the deterministic equation (Eqs. (1-3)). At an intermediate noise level, the noise is dominant for a slow-growth state, and

iii). timescale separation between the epigenetic variable and other variables.
First of all, the change in the E t value must be quickly reflected in the concentration of the target protein concentration P t . If the dynamics of P t are considerably slower than those of E t , then P t would not reach the value giving rise to the high growth rate, even if E t reached the value corresponding to the fast growth. Additionally, the dynamics of P s have to be faster than those of E t . Otherwise, the reached steady-state value of E t cannot afford a high growth rate.
From Eqs. (1-3), E t and = P n s t ( , ) n exponentially relaxes to its steady state solution with the rate approximately l − , and μ + d n st , respectively, where μ st is the mean growth rate near the steady state. Hence, the relation μ + −  l d n st is the condition for the adaptation mechanism to work.

An analytic exploitation of the mechanism.
Noting the time-scale difference among the variables, we can analyze the conditions for this adaptive growth, by deriving the Fokker-Plank equation of E t . Under the assumption that the number of the proteins is sufficiently large, we obtain the Langevin equations corresponding to Eqs. (1)(2)(3). We then assume that the timescales in the change of P t and P s are sufficiently smaller than that of E t and apply the quasi-steady-state approximation: we solve the steady-state of the deterministic equation of Eqs. (2) and (3), and plug the solutions into H(P s ). The noise terms for P t , P s , and E t are integrated into multiplicative noise to Eq. (1) 15  The parameters are given by β is obtained, as is plotted for several values of θ in Fig. 2(a). The peak of the steady distribution shows a transition from the trivial value corresponding to the stable fixed point value of E t in the noiseless limit to the adaptive value ⁎ E E t t . The average growth rate thus obtained is plotted as a function of σ and θ in Fig. 2(b). Accordingly, a boundary which separates the adaptive and non-adaptive regimes is analytically estimated (see Supplement). The estimate well captures the transition between two regimes. The growth rate remains at the high  www.nature.com/scientificreports www.nature.com/scientificreports/ values even if σ increases further as long as θ is sufficiently large, whereas in the infinitely large noise region, our assumption that the effect of stochasticity can be neglected for P s and P t is no longer valid.
To verify the robustness of the presented results, we confirmed that the adaptation mechanism works independently of the specific form of the growth rate and parameter values. Furthermore, even if the epigenetic modification state changes spontaneously, a fast-growth state is selected and sustained, as long as the modifications triggered by the sensor protein dominates the epigenetic modification. The detailed set-ups and the results are presented in Supplement.
In addition, we tested the reliability of our adaptation mechanism in chemical reaction network systems in which metabolites are converted into growth factors via several enzymatic reactions. In the model, the epigenetic variable is implemented to each enzyme, whereas one sensor protein triggers the changes of all the epigenetic variables. As shown in Fig. 3, the proposed mechanism works also in the chemical reaction network model. Details of the model are given in the Supplement.
The achieved growth rate here is still lower than μ max (e.g., 20% of it), the possible maximal value. Recall, however, that initially the growth rate is ~0, and it is enhanced in the orders of magnitude, just by the fluctuations in the epigenetic modication level. The growth rate is then sustained at a high level. Note also that just a random reaction network is chosen here, without tuning the parameter values, either. Optimizing parameter values or network structures would further increase the growth rate.

Discussion
To sum up, numerical simulations and analytic calculation of our model revealed that the epigenetic regulation and stochasticity of chemical reactions lead to an increase in the growth rate of the cell, starting from the stable attractor with a low growth rate. Even though there is no pre-designed gene-regulation or signal transduction network to achieve the high growth rate, the cell reaches a high-growth state, and stays therein driven by stochastic noise and dilution by cell-volume increase.
If the growth rate is high, the dilution is large, so that the protein concentrations relax fast, and the perturbation by noise has little influence. For the slow-growth state, the protein concentrations fluctuate more under the action of noise, whereas the epigenetic state is frequently modified due to the accumulation of the sensor protein, which enables reaching a fast-growth state. This directional epigenetic change driven by noise may be reminiscent of thermal ratchet [17][18][19][20][21] , but here it is due to noise in the reaction and epigenetic process. Given that this "epigenetic ratchet" does not require any tuned design by evolution, it can explain the robust adaptation observed experimentally 2-4 .
Here we briefly review the key features for the present mechanism to work. (i). Protein concentrations are diluted due to volume growth of the cell. This is simply because the concentration is the number divided by the volume. (ii). The rate of epigenenetic modification increases with the sensor protein level, in a sigmoidal manner. This type of sigmoidal dependence is ubiquitous in biochemical reactions, and can be derived, for example, by assuming that the epigenetic modification takes place with the dissociation-association kinetics between DNA and epigenetic modification protein. Then, as in the Hill equation, the sigmoid function is a natural consequence. (iii). The epigenetic level of the target gene determines the production rate of the target protein. While we assumed that the production rate increases linearly with the epigenetic level for most part in the text, the mechanism works with the Michaelis-Menten-type production rate, and even with the decreasing function of E t (see Supplement). (iv). The concentration of target protein determines the growth rate. Taking into account an experimental report that the expression levels of metabolic proteins are tuned to optimise the resource allocation 22 , the existence of the optimal concentration of the target protein will be reasonable, whereas this assumption can be relaxed as discussed in Supplement. (v). The sensor protein needs to be constitutively produced. This point needs further experimental confirmation in bacteria. In some other organisms, however, it is verefied. For example, in one of the mammalian oxidative stress response systems, Nrf2-Keap1 system, a pluripotent transcription factor Nrf2 is continuously produced. Further, it is suggested that the transcriptional regulation by Nrf2 leads to a chromatin remodeling 23 .