Abstract
Cell fate determination, the process through which cells commit to differentiated states is commonly mediated by gene regulatory motifs with mutually exclusive expression states. The classical deterministic picture for cell fate determination includes bistability and hysteresis, which enables the persistence of the acquired cellular state after withdrawal of the stimulus, ensuring a robust cellular response. However, stochasticity inherent to gene expression dynamics is not compatible with hysteresis, since the stationary solution of the governing Chemical Master Equation does not depend on the initial conditions. We provide a quantitative description of a transient hysteresis phenomenon reconciling experimental evidence of hysteretic behaviour in gene regulatory networks with inherent stochasticity: under sufficiently slow dynamics hysteresis is transient. We quantify this with an estimate of the convergence rate to the equilibrium and introduce a natural landscape capturing system’s evolution that, unlike traditional cell fate potential landscapes, is compatible with coexistence at the microscopic level.
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Introduction
In a deterministic description, binary decision making is attributed to the irreversible state transition between two mutually exclusive stable steady states in response to a signal. This state transition is usually governed by regulatory motifs with the capacity for bistability and hysteresis^{1}, thus ensuring that the system does not switch back immediately when the signal is removed^{2}.
The stochastic dynamic behavior of a gene regulatory network is governed by a chemical master equation (CME), which describes the time evolution of the probability distribution of the system state. The stationary solution of the CME is unique and independent on the initial state of the system^{3} and therefore, incompatible with memory effects or hysteresis. The incompatibility of hysteresis with intrinsic noise in gene regulatory networks has been addressed, for example, by Lestas et al.^{4}. However, there are numerous works providing experimental evidence of hysteretic behavior under significant levels of stochasticity^{5,6,7,8}.
In the context of phenotypic switching and cell fate determination, three different scenarios have been distinguished and experimentally observed for binary decision making: deterministic irreversible^{9,10,11}, stochastic reversible^{12}, and stochastic yet irreversible state transitioning^{13}. Reversibility is understood here as the capacity of individual cells to switch back in absence of external signals. According to a pseudopotential interpretation, dynamics are directed by a pseudopotential landscape divided by a separatrix into two basins of attraction such that each local minimum corresponds to a specific cellular state. Stochastic irreversible transitions are found to appear when cells are initialized on (or near) the separatrix^{13}.
In this article we provide a quantitative description of hysteresis and apparent irreversibility in stochastic gene regulatory networks at the single cell level as transient effects, which disappear at the stationary state. Our analysis is based on an accurate approximation of the CME. This means that our results are valid for purely stochastic regimes far from the thermodynamic limit, and thus complementary to those based on the classical linear noise approximation for systems closer to the thermodynamic limit^{4,14}. Since the stationary solution of the CME is unique^{3}, if the solution corresponds to a bimodal distribution, state transitions at the single cells level occur necessarily in a random and spontaneous manner, switching back and forth between regions of high probability.
Fang et al.^{15} experimentally determined an energy potentiallike landscape as the negative logarithm of the probability distribution, as well as the transition rates, based on previous theoretical studies^{16}. In this contribution, we provide a theoretical basis that explains coexistence of different expression states. In fact, under the assumption of protein bursting^{17}, we propose an efficient form of the CME^{17,18} that allows us to construct a meaningful probability based landscape. Furthermore, a clear link between the characteristic kinetic parameters of regulation dynamics and the resulting landscape is established.
Results
Deterministic description
We consider the simplest gene regulatory motif exhibiting hysteresis, a single gene with positive selfregulation (see Supplementary Fig. 1). In its deterministic description, the evolution of the amounts of mRNA and protein X (\(m\) and \(x\), respectively) for the selfregulatory gene network is given by the set of ODEs:
where \({\gamma }_{m}\) and \({\gamma }_{x}\) are the mRNA and protein degradation rates, respectively. \({k}_{m}c(x)\) is the transcription rate, that is essentially proportional to the input function \(c(x)\) which collects the expression from the activated and inactivated promoter states. This function incorporates the effect of protein selfregulation and takes the form^{19,20}:
with \(\rho (x)\) being a Hill function^{21} that describes the ratio of promoter in the inactive form as a function of bound protein:
The above expression, can be interpreted as the probability of the promoter being in its inactive state, where \(K={k}_{{\rm{off}}}/{k}_{{\rm{on}}}\) is the equilibrium binding constant and \(H\in {\mathbb{Z}}\backslash \{0\}\) is an integer (Hill coefficient) which indicates whether protein X inhibits (\(H\,> \,0\)) or activates (\(H\,<\,0\)) expression. Finally, expression (3) includes basal transcription or leakage with a constant rate \(\varepsilon ={k}_{\varepsilon }/{k}_{m}\) (see Supplementary Fig. 1) typically much smaller than \(1\). The parameters of the Hill function employed along the paper are \(H=7\) (the value taken from To and Maheshri^{22}) and \(K=100\), whereas \(\varepsilon =0.05\). Unless other value is indicated, we use \(a=54\). Assuming that \(mRNA\) degrades faster than protein X we have that \({m}^{* }={k}_{m}c(x)/{\gamma }_{m}\) and model (1) reduces to:
where \(\tau =t{\gamma }_{x}\), \(a={k}_{m}/{\gamma }_{x}\), and \(b={k}_{x}/{\gamma }_{m}\). Along the paper we use the values \({\gamma }_{x}=4\times 1{0}^{4}\) s\({}^{1}\) and \({\gamma }_{m}=20{\gamma }_{x}\) s\({}^{1}\), taken from Friedman et al.^{23}.
The selfregulatory network described by the deterministic Eqs. (1) and (2) shows bistability and hysteresis (see Fig. 1a). For a range of the control parameter \(b\) the system evolves toward one stable state or another depending on the initial conditions. We therefore say that the system has memory, since steady state values provide information about the system’s past. In systems with hysteresis (dependency of the state of the system on its past), forward and reverse induction experiments follow different paths resulting in a hysteresis loop (the system switches back and forth for different values of the control parameter)^{24}.
Stochastic description
Gene expression is inherently stochastic. Taking into account that mRNA degrades faster than protein X in most prokaryotic and eukaryotic organisms^{25}, protein is assumed to be produced in bursts^{19,20,23,26} at a frequency \(a={k}_{m}/{\gamma }_{x}\) (see Eq. (5)). From this assumption, it follows^{20,23} that the temporal evolution of the associated probability density function \(p:{{\mathbb{R}}}_{+}\times {{\mathbb{R}}}_{+}\to {{\mathbb{R}}}_{+}\) can be described by a partial integrodifferential equation (PIDE) of the form:
where \(x\) and \(\tau\) correspond with the amount of protein and dimensionless time, respectively. The latter variable is associated to the time scale of the protein degradation, as in the previous deterministic description. In addition, \(\omega (xy)\) is the conditional probability for protein level to jump from a state \(y\) to a state \(x\) after a burst, which is proportional to
with \(b\), as in Eq. (5), representing the burst size. The stationary form of the one dimensional Eq. (6) has analytical solution^{19,20} \({p}^{* }(x)=C{\left[\rho (x)\right]}^{\frac{a(1\varepsilon )}{H}}{x}^{(1a\varepsilon )}{e}^{\frac{x}{b}}\) where \(\rho (x)\) is defined in (4) and \(C\) is a normalizing constant such that \(\int _{0}^{\infty }{p}^{* }(x)=1\). It has been shown that the equilibrium solution associated to a CME is unique and stable^{3}. This is also the case for the Friedman Eq. (6) whose stability has been recently proved by entropy methods^{27,28}, which eventually makes it to qualify as a master equation itself. It is important to remark that stability properties remain valid for higher dimensions (i.e., multiple genes and proteins). While the mean \(x\)values of the stationary solution do not depend on the initial conditions, the means obtained at the transients depend on the initial number of proteins (Fig. 1b).
Note that under sufficiently slow dynamics, transient values may look stationary, thus leading to plots (red and blue lines) that resemble hysteresis, as different mean values coexist within a given interval of the \(b\) parameter. Interestingly, this interval coincides with bimodal distributions in which the two most probable states are separated by a region, in the protein space, with very low probability. This explains recent experimental observations^{29} in which the range of apparent hysteresis was found to shrink with time. Here we denote this phenomenon as transient hysteresis and show how, in fact, the low probability region acts as a barrier that hinders transitions between low and high protein expression, contributing in this way to slow down the dynamics toward the corresponding stationary distribution. Supplementary Fig. 2 compares transient and stationary distributions for different values of the control parameter and different initial conditions. This figure provides a clear illustration of how, in presence of stochasticity, hysteresis is transitory: it shrinks with time and disappears as the system achieves the stationary state.
In order to compute an estimate of the convergence rate to equilibrium we make use of entropy methods^{28,27} and define the entropy norm as \(G= \int _{0}^{\infty } H(u(\tau ,x)){p}^{* }(x){\rm{d}}x\) where \(H(u(\tau ,x))\) is a convex function in \(u\), that in this study has been chosen to be \(H(u)={u}^{2}1\), with \(u=p(\tau ,x)/{p}^{* }(x)\). According to Pájaro et al.^{28} and Cañizo et al.^{27}, \(G\) satisfies the following differential inequality:
where \(\eta\) is a positive constant (its dimension is the inverse of time) related to regulation (parameters \(H\) and \(K\)), as well as the transcriptiontranslation kinetics (\(a\), \(b\)). The smaller \(\eta\), the slower its convergence toward the corresponding equilibrium solution. Computing \(\eta\) requires a full simulation of (6) until the system reaches the equilibrium distribution for each parameter on a given range, what is computationally involved. In this work, the PIDE model (6) is solved by using the semilagrangian method implemented in the toolbox SELANSI^{18}.
Alternatively, we provide a truncation method to approximate the rate of convergence that we use here for verification purposes. The method makes use of the discrete jump process representation (see Supplementary Fig. 3), which is a precursor of Friedman PIDE model, by making the protein amount a continuous variable^{17}. With this method (see Methods section) we compute the negative eigenvalue with smallest absolute value of the state change matrix \({\mathcal{M}}\) which we refer to as \({\lambda}_{1}\). This eigenvalue is a good approximation of the convergence rate \(\eta\).
Figure 2 compares the eigenvalue \({\lambda }_{1}\) with the convergence rate \(\eta\) obtained by simulation, for different values of the parameter \(b\). In the parameter range where bimodal distributions occur, the negative eigenvalue \({\lambda }_{1}\) is a good approximation of the convergence rate of the PIDE model. The figure also shows how the smaller \(\eta\) values correspond to the solution near equilibrium which lies within the hysteresis region in the \(b\) parameter space. Remarkably, low convergence rates coincide with the parameter region in which bimodal behavior take place.
The estimation of the convergence rate (either in terms of \(\eta\) or \({\lambda }_{1}\)) can be obtained from kinetic coefficients \(a\) and \(b\) previously estimated from experiments. To that purpose, we can use the PIDE model to find by least squares from typically time dependent distributions obtained from a cell population by flow cytometry, the best set of parameters. Alternatively, distributions could be reconstructed from single cell time series. With the resulting model, simulations will be executed to estimate rate of convergence.
This example has served as a proof of concept to clarify how hysteresis, as it is known in deterministic nonlinear systems (i.e., as a longterm stationary phenomenon) has not an equivalence in a microscopic world governed by a CME. For stochastic systems, hysteretic behavior is a transitory phenomenon, i.e., it can be only obtained under transients that may resemble stationary solutions due to the extremely slow dynamics at which bimodal distributions evolve. Nonetheless, some correspondence can be drawn between the most frequently visited states on a microscopic system and the stable states on the deterministic counterpart (see Supplementary Note 1).
Figure 3 shows that the logarithm of the eigenvalue decreases as the parameters \(a\) and \(b\) become higher and smaller, respectively. Variations of the logarithm of the eigenvalue are more pronounced inside the bimodal (if one of the peaks lies at zero the bimodal distribution is also known as binary) and bistable regions. Moreover, as discussed by Pájaro et al.^{30}, as the parameter \(a\) increases the system approaches the thermodynamic limit.
Hysteresis in a mutual repression gene network
We consider the gene regulatory network in Ellis et al.^{31} and Wu et al.^{13}, where the Lacl promoter is repressed by the protein expressed by the TetR promoter and vice versa, and ATc is used to inhibit the expression of TetR (see details in Supplementary Note 2).
We simulate the dynamics from two different initial conditions, \({p}_{0}={\mathcal{N}}([600,10],5{\rm{I}})\) and \({p}_{0}={\mathcal{N}}([50,200],5{\rm{I}})\), and take snapshots at 50, 100, and 150 h. In Fig. 4, we depict the dose–response curves at \(t=100\) h for each initial condition (red and blue lines, respectively) and the stationary dose–response curve. It can be observed clearly how hysteresis disappears at the stationary. Note that the transient hysteresis observed at \(t=100\) h is in agreement with experimental observations by Wu et al.^{13}.
The transient distributions are depicted in Supplementary Figs. 4 and 5 representing the corresponding marginal distribution for the same snapshots. As it is shown, the distribution at 50 h resembles a stationary distribution, since no significant differences are observed with those obtained at \(t=100\) h and even at \(t=150\) h. However, comparing those distributions with the stationary distribution (see also third row in Supplementary Fig. 6), we clearly conclude that the system is not at the stationary state. Thus, the corresponding dose–response curve at \(t=50\) h describes a transient hysteresis phenomenon. Note that as shown in Supplementary Figs. 6 and 7 even snapshots taken at much longer times (e.g., 1500 h) still differ significantly from the stationary solution.
The results for \(t=50\) h are coherent with the observation by Wu et al.^{13} that if a trajectory starts clearly within one of the basins of attraction remains there for a long time. Note that the time needed to reach the stationary state might be longer than the natural timescales of relevance to the process. This is in accordance with Wu et al.^{13} where the transitions are characterized as stochastic yet irreversible.
The convergence rates of the solution toward the equilibrium are depicted in Fig. 5. As it happens for the 1D example in Fig. 2 the parameter region leading to bimodal distributions corresponds with the slowest convergence rates; such slow dynamics is responsible for the phenomenon of transient hysteresis.
Supplementary Fig. 8 compares the set of stable and unstable equilibrium states obtained from a deterministic representation with the most and least probable microscopic states, respectively. Note that this equivalence does not support the existence of longterm (stationary) hysteresis at the microscopic level. Essentially, what the picture shows is that, rather than a parameterdependent preferential state among two stable ones, there are two highly probable states that coexist for a given parameter region on a cell population.
Discussion
These results provide us with an important insight on how to interpret experimental results showing hysteretic behavior at the level of gene regulatory networks: if the system is governed by the CME, hysteresis is necessarily transient. Note that for slow dynamics (high \(a\) and low \(b\) values) the time needed to reach the stationary state might be longer than the natural timescales of relevance to the process. This is in accordance with previous studies reporting large mean passage times^{14} and also with Wu et al.^{13} where they engineer a synthetic switch with stochastic yet irreversible transitions (the same mutually inhibitory gene regulatory motif is analyzed next using our PIDE approach).
The characterization of a cell response as hysteretic or nonhysteretic is important. For example, in a a recent study concerning epithelial to mesenchymal transition (EMT), a process through which epithelial cells transdifferentiate into a mesenchymal cell fate, the authors characterize two types of responses, hysteretic and nonhysteretic EMT, and report the notable influence of hysteresis on the metastatic ability of cancer cells^{32}.
Invoking pseudopotential concepts to interpret dynamics in GRN under fluctuations^{13}, although attractive from an intuitive point of view, may be misleading since it cannot capture the notion of coexistence. By coexistence we mean that two different protein expression levels coinciding with the peaks of the bimodal distribution coexist on a cell population (assuming no cell to cell variability on the initial conditions).
The pseudopotential landscape is not easy to compute either, specially when increasing the number of proteins expressed. Alternatively, we can use the stationary solution of (6) to construct on the natural framework of probability distributions, a landscape informing of the possible transitions or evolution of the underlying microscopic system. As we illustrate in the example discussed in the supplementary material, its computation can be extended in a straightforward manner to larger dimensional protein spaces. This can be of use to efficiently identify most prevalent phenotypes coexisting on a given cell population.
The main assumption of the PIDE model is protein bursting (mRNA degrading faster than proteins). As reported in Pájaro et al.^{17}, the approximation remains generally valid even for degradation rate ratios around 2–3 (5 in the most restrictive cases). In terms of protein copy numbers, although the PIDE model is valid in any range, we expect a significant effect of the inherent intrinsic noise for low copy numbers (in the order of thousands and lower). Note that, for prokaryotic cells this is the case for the majority of the proteins^{33}. Although in eukaryotic cells proteins are in general more abundant there is still a significant portion of the cases for which the copy numbers appear to be low (see for example, Schwanhausser et al.^{34}, Shi et al.^{35}, Nguyen et al.^{36}). We would like to remark that extrinsic noise is not taken into account in this study since we are quantifying the effect of intrinsic noise in hysteresis.
Methods
Stochastic model and simulation
We use the PIDE model^{17} described in Eq. (6). The model is simulated by a semilagrangian method implemented in the toolbox SELANSI^{18}.
Rates of convergence (truncation method)
Let \({\mathcal{P}}:{{\mathbb{R}}}_{+}\times {\mathbb{N}}\to [0,1]\), be the probability of having \(n\) proteins at time \(\tau ={\gamma }_{x}t\). The time evolution of \({\mathcal{P}}(\tau ,n)\) is given by the following CME with jumps that reads
where the transition probability \({g}_{i}^{j}\) is proportional to the production rate of messenger RNA, so that
In order to obtain an approximation of the convergence rate of the PIDE model toward the stationary state, we use the truncated form of the discrete Eq. (9). Let \(N\) be the maximum possible number of proteins. Then, Eq. (9) can be written in matrix form as
where the matrix \({\mathcal{M}}\) reads
with the elements of the diagonal \({d}_{i}\) being of the form
equivalently
The steady state is given by the null space of matrix \({\mathcal{M}}\), which is spanned by the normalized eigenvector associated to the unique zero eigenvalue, as the associated eigenspace has dimension one. Actually, since the graph associated to matrix \({\mathcal{M}}\) (See Supplementary Fig. 3) has one trap, all the eigenvalues are negative except one (which is zero)^{37}. By \({\lambda }_{1}\), we denote the negative eigenvalue closer to zero, i.e., the one with smallest absolute value.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
Data availability
All relevant data needed to reproduce the results are included in the text and supplementary information.
Code availability
The semilagrangian method to simulate the PIDE model is freely avaliable and can be downloaded at: https://github.com/selansi/Selansi.
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Acknowledgements
M.P. and A.A.A. acknowledge funding from grant PIE201870E041; I.O.M. acknowledges funding from Spanish MINECO (and the European Regional Development Fund) project SYNBIOCONTROL (grant number DPI201782896C22R). C.V. has been partially funded by the spanish MINECO project MTM201676497R and Xunta de Galicia grant ED431C2018/033.
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A.A.A. and I.O.M. conceived the research. M.P. and A.A.A. performed the research. C.V., M.P., I.O.M., and A.A.A. contributed to the simulation methods. A.A.A., M.P., I.O.M., and C.V. wrote the paper. A.A.A. supervised the project.
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Pájaro, M., OteroMuras, I., Vázquez, C. et al. Transient hysteresis and inherent stochasticity in gene regulatory networks. Nat Commun 10, 4581 (2019). https://doi.org/10.1038/s4146701912344w
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DOI: https://doi.org/10.1038/s4146701912344w
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