Classical Methods and Issues

The deterministic mean-field description of this dimer model through quasi-steady state approximation (QSSA) yields the ODEs.

where , and are the mean number of the mRNA, protein and dimer respectively, and the parameters are shown in Fig. 1. This system has two stable fixed points and one saddle in physically reasonable parameter regime (see Text S1:I). These two stable fixed points correspond to the expressed and unexpressed states at which the copy number of proteins is at high or low state, respectively. With this deterministic description, once the system settles in one of its two attractive fixed points, it will stay there forever. However, in the presence of intrinsic noise, the system will fluctuate around its attractive fixed points and switch between these two metastable states on a large timescale. This type of switching among metastable states under small perturbations is exactly the rare events studied in the literature. The large deviation theory (LDT) is an appropriate tool to quantitatively describe the rare transitions [23], [25], [27]. Roughly it tells that when the system size is sufficiently large, the probability that the trajectory of the stochastic dynamics stays in a small -neighborhood around a specific path can be given as (1)where is called the rate functional. Thus the most probable transition path can be obtained by minimizing associated with the Lagrangian the . Our task is to find the for specific models. For Gillespie type birth-death dynamics, has no closed form and only its dual Hamiltonian can be obtained in the large volume limit , i.e. the number of all types of molecules goes to infinity. However, this approach encounters difficulty if we take the DNA switching into consideration since there is only one DNA copy in the considered model. Thus the straightforward utilization of the existed Hamiltonian in the large volume limit is invalid here.

Large Deviation Theory

To solve this issue, we develop the LDT directly for this specific system following the way in [28] with further extension. The biologically relevant choice of parameters in our model suggests the scaling , and the others are , where is the system size which is usually chosen as the typical number of proteins in the expressed state. This is also the correct scaling under which the mean field limit of the CMEs gives the ODE system derived from QSSA (see Text S1:I). Define the rescaled concentration variable where and is the state vector for the number of mRNAs, proteins and dimers. Correspondingly define and to transform all the parameters to magnitude. However for notational simplicity we will drop the tilde symbol on these parameters in the rest of the paper. It turns out that the Lagrangian of our model has the form.(2)which combines the LDT result for large volume limit as shown in (1) and the Donsker-Varadhan type LDT result [22], [29] for DNA fast switching. Here resembles the velocity in classical mechanics, is a probabilistic 2-vector describes the residence distribution of DNA at the inactive or active state.

Let us illustrate the construction of (2) via an intuitive way as follows. The net effect of the fast switching of DNA induces a residence distribution with components and characterizing the probability of DNA staying at the inactive and active states, respectively. Whenever DNA is at the inactive or active state, we can apply the traditional LDT result in the large volume limit. This leads to the first part in (2). The second part describes how the visiting distribution induced by the random fast switching of DNA is close to a prescribed residence distribution given the current state . This is exactly what the Donsker-Varadhan type LDT gives [22], [29], which is described by . Since we are only interested in the LDT for the state variable , the overall Lagrangian should be taken infimum with respect to all of the possible residence distributions . All of the statements will be made clear in the continued paragraphs.

Similar as the case in the large volume limit, it is not feasible to get the explicit form of the Lagrangian in general, but its dual Hamiltonian can be available, where is the generalized momentum conjugate to as in classical mechanics. They are connected via the Legendre transform.

(3)

For the Gillespie type birth-death process with reaction channels and the propensity function and stoichiometric vector for , the existed LDT result gives the Hamiltonian [23].(4)

Specifically in our model we have.(5)with the Hamiltonian(6)when the DNA is at the inactive state and(7)when the DNA is at the active state. Here and corresponds to the part of the Hamiltonian for the gene expression, i.e. the transcription and translation processes. On the other hand, the famous Donsker-Varadhan LDT gives the Lagrangian(8)where is any 2-vector and is the infinitesimal generator for the DNA two-states jumping process at a given state defined as

(9)The direct calculation shows that(10)

Combining Eqs. (3), (5), (6), (7) and (10), we obtain the final explicit Hamiltonian by optimization.(11)where and . This derivation can be easily extended to similar problems.

It is worth noting that one can show the Hessian of with respect to has the form.(12)where , and Since are positive when are positive, the decomposition in (12) shows that the Hessian is positive definite. This means the Hamiltonian is convex with respect to in physically meaningful domain. It is quite different from that obtained by WKB asymptotics [15] (See Text S1: II). The convexity of the Hamiltonian is testified to be essential for the robustness and efficiency of the numerical algorithm [25] both theoretically and practically. It can be rigorously proved that it is a natural by-product from the LDT analysis.

Quasi-Potential and Optimal Transition Path

With the obtained LDT, we can get the optimal transition path through variational optimization. Moreover, we can define the local quasi-potential with respect to a meta-stable state as.(13)

From classical mechanics, the local quasi-potential satisfies a steady-state Hamilton-Jacobi equation characterized by the Hamiltonian shown in Equation (11).(14)

The LDT also confirms the equilibrium distribution of the system through a global quasi-potential function .(15)where the symbol means the equality relation on a logarithmic scale, and can be obtained from its local version by a suitable sticking procedure which we will describe in the next subsection. This function , which naturally serves as a rationalized version of the Waddington potential, is one main point of this article.

The classical Hamilton-Jacobi theory enables one to solve the local quasi-potential satisfying (14) with variational methods. Here we employ the powerful geometric minimum action method (gMAM) proposed in [25] to compute by minimizing the action functional with a prescribed Hamiltonian (11). The key idea of gMAM is essentially the Maupertuis principle in classical mechanics, which reformulates the action functional on the space of curves with intrinsic parameter, thus frees the time variable in the minimization process and still keeps its efficiency in high dimensions. This approach also resolves the issue of the singular boundary value problem by solving Hamilton’s equations directly [15], [30]. Specifically after each run of gMAM with fixed starting and ending points, one obtains the minimized action and the corresponding optimal path. The readers may be referred to Text S1:IV for more details.

It is worth asking whether the choice of the large parameter affects the final results since any choice is artificial in practice. An affirmative answer is given in Text S1:III that only the scaling matters and the final systems are equivalent with respect to different choices of the large parameter .

Construction of Global Quasi-Potential Energy Landscape

Based on the obtained local quasi-potential starting from the on and off states, we may construct the global quasi-potential energy landscape for genetic switching model by sticking them together. The system with only two metastable states and one saddle point, as our dimer model, can be handled conveniently as the way shown below. The readers may refer to [24] for systematic methods of sticking the global quasi-potential for more complex systems.

In our dimer model, we first compute the local quasi-potential starting from two metastable states and . We define and , where is the saddle point. Denote . Suppose , then the global quasi-potential is given by.(16)otherwise has the form

(17)It is not difficult to observe that sticking the two local quasi-potential via the linking saddle is the key point in this construction.

In most cases, the considered system is in high dimensions while we are only interested in partial variables which is in low dimensions. This is also the case in our dimer model. Although the global quasi-potential is in three dimensions, we are mainly interested in its 2D configuration in the mRNA-Protein plane. So we need to reduce the redundant dimension to obtain a 2-D potential . We proceed with the following arguments.

According to the LDT analysis (15), we obtain.(18)With the same reason, we have(19)where is the reduced distribution for mRNA and protein. By definition, this distribution is given by(20)From the Laplace asymptotics [29] we get a simple reduction strategy.(21)

This argument is general for any high dimensional situations and indeed it is also applied to the noise cascading model considered in our later text.

Optimal Transition Path

The large-deviation theory predicts that when events with little likelihood occurs, they will follow the optimal transition path which minimizes the action (13) with high probability. The probability of those paths deviated from the optimal one will decay exponentially. By choosing two stable states as the starting and ending points respectively, we obtained the switching path from either of the two states (see Fig. 2). For the convenience of visualization, we project the transition paths onto the mRNA-protein plane.

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Figure 2. Switching paths (A) from off to on state (purple solid curve) and (B) from on to off state (red solid curve) and MC simulations for both switching trajectories.

We take the two stable fixed points in the deterministic dynamics as the starting and ending points. Darkness of the shading points represents the number of visits for reactive trajectories with smoothing. (C) Averaged switching trajectories from MC simulation. For each number of protein, we average in the mRNA dimension using probability as weight. Here the statistical results around each stable state is not shown because of the restrictions by our MC simulation algorithm (see Text SI:VI-A). The results are obtained from 1000 independent long time MC simulations. The parameters here are , , , , , , , , and

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Figure 2 shows clearly that when switch occurs, the trajectory prefers to be around the most probable path characterized by the Hamiltonian (11). The fact that the off-to-on and on-to-off paths are not identical agrees with the previous studies that the switching process is irreversible. The irreversibility is fundamental in chemical reaction kinetics due to the non-gradient nature of the considered system and can be considered as a form of hysteresis. However, in contrast with the previous study [11], our results indicate that when the noise level goes to zero, both optimal transition paths pass through the same bottleneck, i.e. the saddle point obtained from the corresponding deterministic model. This suggests that the saddle point has the lowest barrier height along its stable manifold and is in accordance with the energy landscape shown later.

Global Quasi-potential Energy Landscape

Applying the method of constructing global quasi-potential, we can compute the 2-D potential for our dimer model. The result is shown in Fig. 3.

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Figure 3. Quasipotential energy landscape of the whole genetic switching system with (A) two and (B) three dimensional view as well as switching paths between two stable fixed points.

Each path passes through the saddle point. Here, the parameters are the same as in Fig. 2.

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In Fig. 3, we observe that the on and off states correspond to two local minimum on the quasi-potential energy landscape, the saddle of the deterministic dynamical system exactly corresponds to the saddle point on the quasi-potential energy landscape too. The flatness along the mRNA direction keeps in good accordance with the large fluctuation observed in the reactive trajectories.

To further characterize the switching path, we denote the first half (i.e. the part between the starting point and the saddle point) as the uphill path and the latter half as the downhill path. One may note that the transition path is also given by the Hamilton’s equations , . Therefore based on the fact , we obtained when . At the saddle point in any transition path, we have [25], and thus along the whole downhill path. With this result we obtain the downhill equations , which exactly corresponds to the corresponding deterministic dynamics. This fact explains that after climbing the saddle point the biological system relaxes to its attracting state fast without costing any action. This fact was also pointed out in [30]–[32].

On the other hand, the Hamilton-Jacobi theory also yields the uphill dynamics.(22)

It is difficult to give a thorough understanding about the whole uphill path because of the general nonlinearity of . However, an analysis around the critical points is instructive. Based on the fact at critical points (i.e. the metastable states and saddle), we have by Taylor expansion.(23)

Recall that corresponds to the deterministic mean field ODEs, the equation (23) is exactly the uphill path of a chemical Langevin dynamics [33] (See more details in Text S1:VI). However, this chemical Langevin dynamics is not a straightforward generalization from the mean field ODEs like the usual large volume limit. Indeed, this reflects the specialty in our model setup which is related to the DNA fast switching. To see this more concretely, we have the approximated uphill dynamics for the -component in our dimer model as.(24)

The corresponding effective Langevin dynamics for the -component reads.(25)where and are two standard temporal Gaussian white noise with mean and covariance , and It is remarkable that besides the mean field transcription rate , we have an additional term in . This is quite different from the usual chemical Langevin equations where for the -th reaction the diffusion term has the same form as its corresponding drift part except a square root operation. The additional term in makes that the fluctuation of the transcription is larger than that in gene expression processes without DNA switching, yet has the same mean field transcription rate This observation coincides with the theoretical analysis in [34] although no feedback is considered there.

The quasi-potential energy landscape not only provides the pictorial illustration for the dynamical transitions, it also contains many quantitative information to understand the metastability in genetic switching models. Once the global energy landscape is obtained, one can get the stationary distribution of the whole system via Eq. (15). Furthermore, it is very easy to calculate two main characteristic quantities used to describe a genetic switching system through only small amount of computational efforts. One is the transition rates, corresponding to the lowest barrier heights between two metastable states, and the other is the noise strength, corresponding to the steepness of quasi-potential around each metastable state.

Global Property: Mean Switching Time (MST)

The transition rate of switching systems is often characterized by Mean Switching Time (MST). We can compute the MST from either metastable state in the dimer model. For example, according to [24], the MST from on-to-off transition can be estimated from an asymptotic analysis.(26)

Here is a prefactor, and the quasi-potential energy barrier is , where and are the action values at the saddle and on states, respectively. The result for the MST of off-to-on transition is similar. Although for one dimensional system the prefactor of MST can be obtained [31], there are no available results in high dimensions because of the geometry problem and the non-gradient nature of the system [35], [36]. Fortunately, the prefactor varies slowly in many cases, therefore we can compare the MC simulations with the exponential time part and adjust the prefactor to fit the numerical results.

The sensitivity of both MSTs, and , to the change of transcription rate and mRNA decay rate are investigated and compared with MC simulations in Fig. 4A and Fig. 4B, where the prefactors of off-to-on and on-to-off transitions are estimated as 2300 and 29, respectively. It can be observed that the MST is excellently predicted by Eq. (26) up to a slowly varying prefactor. And it is worth noting that when the MST becomes very large, the efficiency of the classical MC simulations gets extremely low while it is well kept in our approach.

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Figure 4. The mean switching time (MST) and quasipotential energy landscape as a function of parameters.

(A) and (B): MST as a function of transcription rate . Promoter transition rates , the gMAM results with numerical prefactor of off-to-on transition (red solid line) and on-to-off transition (blue dashed line), compared with MC simulations () and (), respectively. (C) and (D): The gMAM results with different promoter transition rates of off-to-on transition (red) and on-to-off transition (blue), where solid line with is same as (A) and (B), the faster transition rate in dashed line with , the slower transition rate in dotted line with . Other parameters are ; in (A,C), and (B,D)

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The positive feedbacks in genetic circuit usually provide cellular memory or all-or-none switch. The results in Fig. 4 reveal the robustness and sensitivity of the dimer model to the change of different kinetic parameters. In Fig. 4A and Fig. 4B, the promoter transition rates . When the transcription rate increases from 360 to 420, the MST from on-to-off states increases exponentially, while the MST of off-to-on transition decreases slowly; it means the on-state becomes more stable while the off-state can still keep its stability. Therefore when the transcription rate is increased, our genetic dimer circuit with positive feedback provides a stable cellular memory at the on state, but the off state remains its stability. Thus the system can not switch from the off state to the on state effectively.

How to turn on the genetic switch? The results in Fig. 5 provide two possible effective choices. The first choice is to pose an additional source of the mRNA production, which we call as the trigger signal. We denote the additional mRNA production rate as . If increases from 0 to 20, then MST of off-to-on transition will decrease exponentially and the genetic switch is turned on. We also show how the global energy landscape changes with different trigger rate in Fig. 5C and in Fig. 5D. It is evident to see from the figures that when the trigger signal increases, the barrier height from off-state to on-state decreases, which is in accordance with the MST of off-to-on transition . The relevant biological switch can be found in the start point of budding yeast cell cycle process, where the additional trigger signal in G1 cyclin Cln3 causes the activation of G1 transcription factor SBF and MBF [37]. The second possible choice is to decrease the degradation rate of protein . When the degradation rate of protein decreases from 1 to 0.5 in Fig. 5B, MST from off-to-on transition will also decrease exponentially from to and turn on the genetic switch. This is the case in Bacillus subtilis, where the transitions into competent state is caused by decreasing the degradation rate of protein ComK [38]. All the other relative parameters are listed in the caption of Figures.

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Figure 5. The mean switching time (MST) of off-to-on transition as a function of (A) trigger signal strength that transcribes mRNA at constant rate and (B) degradation rate of protein .

(C) and (D): Quasipotential energy landscape with different trigger strength. in (C), and in (D). Other parameters are ; in (A,C,D), and in (B).

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Furthermore, we calculate the MST of both from off to on and on to off states in the different promoter transition rates. We show the results with the fast rates () in Fig. 4C and slow rates () in Fig. 4D. These results indicate that the system with slow promoter transition rates tend to have short MST of off-to-on transition (red dashed lines) while fast rates lead to long MST of off-to-on transition (red dash-dotted lines). This is due to the reason that faster promoter transition rates lead to smaller mRNA and protein noise strength (see Fig. 6 for more detailed information). Ignoring the difference of the mechanism of initial transcription between prokaryotes and eukaryotes, in the simple case, the faster promoter transition rates correspond to the gene expression process in prokaryotes, and the slower promoter transition rates correspond to the slow chromatin remodeling process in eukaryotic case [39]. The results suggest that prokaryotes may have stronger cellular memory than eukaryotes.

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Figure 6. The coefficient of variation (CV) versus mean number of (A) mRNA and (B) protein induced by varying transcription rate with different promoter transition rates.

The lines and discrete dots correspond to analytical results and MC simulations, respectively. The results with fast promoter transition rates are shown in blue dash-dotted line and , medium rates in red solid line and , and slow rates in magenta dashed line and ◊. The parameters here are the same as in Fig. 4 (A,C).

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Local Property: Fluctuation Around Stable States

Another quantitative information that quasi-potential energy landscape can provide is the noise strength of stable states. Here we use the coefficient of variation (CV, i.e. the standard deviation over the mean) to measure the strength of fluctuation instead of the Fano factor, for the system here has positive feedback thus deviates far from Poisson statistics. Notice that the stationary distribution , we can expand in the vicinity of high stable state up to second order thus get the Gaussian approximation.(27)

Here, , , and is the determinant of matrix . Eq. (27) holds only in the vicinity of the on state with standard deviations and . With the and above, we can easily obtain the CV as shown in Fig. 6.

Figure 6 demonstrates that when the average expression levels increase, the noise strength of mRNA and protein decreases in our positive feedback model. The fluctuation of mRNA is usually larger than that of protein. Furthermore, the noise level with slow promoter transition rates is almost always larger than the one with fast promoter transition rates. This is in accordance with the results of MST that the system with long MST has small noise and vise versa. The inconsistent portion between analytical and simulation results (the left part of the line with slow promoter transition rates in Fig. 6B) is due to the inapplicability of Eq. (27) during the low barrier crossing process for the on state. More details may be referred to Text S1:V.

Application in Transcriptional Cascades

To further illustrate the power of quasi-potential energy landscape and the abundant quantitative information it contains, we apply our methodology to a transcriptional cascades model based on the previous work of S. Hooshangi et al. [40]. In their work, S. Hooshangi et al. synthesized transcriptional cascades comprised of one, two, and three repression layers and analyzed the sensitivity and noise propagation as a function of network complexity. They used different concentrations of anhydrotetracycline (aTc) as inducer and measured the fluorescence intensities of protein eyfp (the last layer of each cascade) by the flow cytometer.

Here we simplify the 3-layer cascades as , where denotes the concentration of aTc as inducing signal and denote the output of proteins in different layers respectively. Then we directly construct the quasi-potential energy landscape for each layer and obtained the normalized probability distribution of the output to certain signal from Eq. (15). The dose response curves to increasing signal are shown in Fig. 7, which are consistent well with the previous experimental results. Further more, two features of transcriptional cascades can be observed. Firstly, the more layers the transcriptional cascades have, the sharper the response curves are (as the Hill coefficient of the 3-layer cascades is 2.00, 3.15 and 4.08 respectively). Thus the sensitivity is increased in the cascades. Secondly, the fluctuation of output can be described by the spreading width of its distribution, so more layers of cascades amplify the cell-cell variability (see Fig. S2). In short, when a cascade has more layers, its response curve gets steeper with a wider probability distribution and thus larger fluctuations. The straightforward calculation of CV based on Fig. 7 has been done and it agrees well with the MC simulations (see Text S1:VI and Figure S2).

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Figure 7. The dose response curves and probability distribution of the output protein in the 3-layer cascades (denoted by ) as a function of inducing signal , 1-layer in (A), 2-layer (B) and 3-layer (C).

The probability distribution can be directly obtained from Eq. (15) after normalization. The Hill coefficient for each cascade is fitted as 2.00, 3.15 and 4.08 respectively.

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Limitations of The Study, Open Questions, and Future Works

We have already illustrated a general methodology based on LDT to quantitatively understand the metastability in gene expression processes perturbed by the intrinsic noise and applied it to a dimer auto-regulatory circuit model. It is clear that this methodology can be extended to more general systems, provide one can explicitly write down the Hamiltonian of the system. If all of the considered species have relatively large numbers, the Hamiltonian is simply the Eq. (4). For the case where the large volume limit fails to be true, our method is also applicable under an additional assumption that the low copy number of species reach their stationary distribution much faster than the others. This is the situation that we treat DNA in our dimer model. However, we would like to mention the limitations of our work, which of course motivates us for future studies.

The main limitations or the corresponding open questions can be summarized into the following three aspects:

1. The case where the large volume limit and the fast switching mechanism are both invalid. This prevents us to construct the LDT for the considered system. Thus there is no Hamiltonian and the current methodology fails. How to quantitatively study such systems and define the proper Waddington energy landscape is an issue.
2. The curse of dimensionality. When the problem is extended to high dimensions, although the computation of optimal transition path and the analysis of MST and CV can be achieved with a reasonable cost, constructing the whole global energy landscape is not feasible in general. However as we have argued before, the whole energy landscape is even not necessary since we are only interested in its configuration for partial components. How to extract these information directly from the Hamiltonian by smart utilization of Eq. (21) is what we are trying to do.
3. Transition rate formula in the high dimensional case. Despite the transition rate formula, i.e. the Arrhenius type formula, for the equilibrium models are well developed [41], there is no complete answer for the non-equilibrium case. The rate with form has long been established in [24], but the prefactor is not known. In one dimensional case, partial result is given [31]. But its high dimensional form is still an open question.

To understand the transition behavior for more general biological systems driven by noise, the above open problems should be overcome in the future studies.