Modeling framework

In order to investigate stability of gene expression patterns without external input gradients, we performed stochastic simulations of a spatial pattern of four mutually repressing genes, using NS-FFS. Here we opted for a minimal spatially resolved stochastic model, shown in schematic Fig 1, inspired by the posterior gap gene pattern in Drosophila development [49]. The model considers four mutually interacting genes A, B, C and D, arranged in a five-stripe pattern (with order A-B-C-D-A) along a cylindrical spatial lattice. The four genes are analogous to the arrangement of the expression domains of gap genes hb, kr, kni and gt in nuclear cycle 14 in the posterior half of the early fly embryo, where hb is expressed in two stripes, in the first (anteriormost) and last (posteriormost) stripe [49–54]; in the following, we use the term “expression domain” or just “domain” of gene A for referring to both of the A stripes together. The spatial lattice consists of N_z × N_ϕ equally spaced and well-stirred reaction volumes with periodic boundary conditions in the circumferential (ϕ-) direction, motivated by the arrangement of cortical nuclei in the developing fly embryo. Protein diffusion and nuclear exchange are modeled via hopping between neighboring reaction volumes, with a rate proportional to the diffusion coefficient. In each nucleus, proteins of the genes A, B, C and D are produced from their corresponding promoters, dimerize and mutually repress each other by promoter binding. Each gene can repress the promoter of each other gene. Repression is non-competitive, i.e., each promoter has binding sites for each of the three other genes’ dimers and is inactivated when at least one dimer is bound (“OR”-logics). The model combines transcription and translation into one production step, neglecting some features of eukaryotic gene expression such as transcriptional bursts and enhancer dynamics, but previous work has shown that this does not alter the results qualitatively [14, 59]. We provide a list of the biochemical equations governing the dynamics of our system in Sec S1.1 and a graphical summary in Fig L in S1 Text.

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Fig 1. Schematic of the spatial gene-regulatory model.

We use a cylindrical lattice of reaction volumes to mimic the arrangement of cortical nuclei in the posterior Drosophila embryo at developmental cycle 14. In each nuclear volume (shaded squares) we simulate production, degradation, dimerization and mutual repression of the four genes A, B, C and D via the Gillespie algorithm. Each gene is subject to repression by the protein dimers of the other genes, as indicated by the schematic promoters. Neighboring nuclei can exchange monomers and dimers via diffusive hopping. The system is initialized in a five-stripe pattern of expression domains in the order A–B–C–D–A, corresponding to the experimentally observed order in the fly embryo. The strength of mutual repression varies among gap gene pairs: genes associated with nearest neighbor (NN) domains repress each other weakly (dashed arrows), while next-nearest neighbor (NNN) domains exhibit strong mutual repression (thick arrows). By default, the concentration of A is pinned at the system boundary where the set of modeled reactions differs from the rest of the system by the fact that the A promoter can not be repressed. See “Materials and methods” section for details.

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In the anterior-posterior arrangement A-B-C-D-A, the genes repress each other mutually via the characteristic pattern of strong next-nearest neighbor (NNN) and weaker nearest-neighbor (NN) repression (alternating cushions), as observed in the Drosophila embryo [43, 49, 54, 60–64]. Specifically, there are two pairs of strongly repressing genes, (A,C) and (B,D), and four pairs of genes that repress each other weakly, (A,B), (B,C), (C,D) and (D,A). In our model, the difference in repression strength is tuned via the unbinding rates of the repressors from the repressed promoter. The strong-repressor unbinding rate is set to a fixed value such that the NNN gene pairs (A,C) and (B,D) are in the bistable regime, while the weak-repressor unbinding rate that tunes the repression between NN gene pairs is varied (ranging from very high unbinding rates, corresponding to very weak repression, towards rates as low as , which also brings the NN repressive interactions into the bistable regime). Here being in the bistable regime means that in individual nuclei only one of the two strongly repressing genes can be expressed at a high level, while its counterpart is expressed at very low level, e.g. the state in which A is expressed at high and C at very low levels, or vice versa. Since the repression between the two genes is assumed to be symmetric, i.e. A and C unbind from each other’s regulatory region with the same rate (and likewise for B and D), these two mutually exclusive stable expression states are equally probable without any further inputs that could break their symmetry, and therefore form a perfectly symmetric “genetic switch”. Thus, in the absence of external cues capable of forcing the bistable systems into a preferred state, stochastic switching is expected to eventually result in one of the domains to dominate over the respective other domain in the NNN pair, causing its elimination and simultaneous expansion of the dominating gene’s domain. This partial breakdown of the initial pattern can happen independently for both strongly repressing NNN pairs and thus in random temporal order; however, ultimately one of the strong repression partners is eliminated in each of the NNN pairs and the system settles in a new, effectively irreversible state in which only the remaining two genes are co-expressed.

On the one hand, we expect that the presence of the third expression domain in between the NNN pair domains can impede elimination of (one of) the NNN pair domains when additional NN repression is present, because it can spatially move apart the strongly repressing (NNN) expression domains and form a “cushion” domain between them, effectively replacing one interface of strong competition by two interfaces of weak competition that allow for local coexistence of the competitors. On the other hand, overly strong NN repression is expected to enhance pattern breakdown because then even the overlapping NN expression domains are brought towards the bistable regime. We therefore study the pattern stability as a function of the repression strength ratio κ, defined as (1) where and are the repressor unbinding rates for weakly repressing NN pairs and strongly repressing NNN pairs, respectively. κ is varied through the weak repression unbinding rate . For κ = 1, i.e. , both the NNN and the NN gene pairs are deeply in the bistable regime and repress each other strongly (because both and are low), while in the opposite limit κ → ∞ () only the NNN gene pairs form bistable switches whereas the NN pairs do not affect each other at all.

Pattern stability is quantified by asymmetry factors

In order to quantify pattern stability, here we define how we understand pattern collapse and construct order parameters that track pattern destruction by mapping the pattern dynamics onto a low-dimensional phase space. A typical “intact” spatial pattern of gene expression with (roughly) equally-sized domains is shown in Fig 2. We consider patterns in which the expression domain of one gene is lost completely as being “destroyed” (note that in our terminology the expression domain of gene A refers to both stripes at the system boundaries). In our system, the strong mutual NNN repression and resulting bistability effectively prohibit coexistence of the strongly repressing genes at one location. Hence, an increase in the size of one domain is always accompanied by a reduction in the size of the domain belonging to the strongly interacting partner. This lead us to introduce the following two order parameters, λ_AC and λ_BD, here termed asymmetry factors, that measure the asymmetry of the expression domain sizes for each of the two strongly antagonistic NNN pairs: (2) Here [P]_tot is the total copy number of P proteins (counting dimers twice), and N = [A]_tot + [B]_tot + [C]_tot + [D]_tot is the total protein number in the system across all species.

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Fig 2. Spatial pattern of gene expression.

Snapshots of the total copy numbers of all considered patterning proteins as a function of the axial coordinate z of the cylinder, averaged over its circumference. Colors correspond to Fig 1 (green = A, blue = B, red = C, black = D). Snapshots were taken every 60 min over a total simulated time of 20 h after an initial relaxation phase of 30 min, starting from rectangular domain profiles of equal length. No-flux boundary condition at either end.

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In the spatially well-ordered pattern each protein domain occupies roughly the same fraction of the system, such that λ_AC ≃ λ_BD ≃ 0.25. As expansion of a domain progresses at the expense of its strong antagonist, λ_AC (or λ_BD) is enlarged and reaches values around 0.5 when the shrinking domain is eventually lost. In order to track progress of the complete pattern losing one of its domains, we use the sum λ = λ_AC + λ_BD, with values around 0.5 for five-stripe patterns and values above 0.75 indicating pattern breakdown.

Our initial simulations revealed that even for very low protein copy numbers (≲ 20) the waiting times until one domain is lost are long compared to the duration of the actual breakdown event, and therefore difficult to sample by direct simulation. This lead us to ask whether the pattern breakdown is merely a slow random process akin to an unbiased random walk in configuration space, or whether the patterns have intrinsic restoring capabilities which would counteract the breakdown process; in such scenario, many more (counteracted) random attempts would be required for concluding the pattern breakdown, effectively rendering it a barrier crossing problem in which the transition states towards destroyed patterns form the barrier. In order to resolve which of these alternative mechanisms is responsible for the stabilization of the expression pattern, we combined our stochastic simulations with Non-Stationary Forward Flux Sampling (NS-FFS), which is particularly suited for enhanced sampling of non-equilibrium rare events. We used λ as the progress coordinate for NS-FFS, which aims at generating a branched and weighted trajectory ensemble that, in the most favorable cases, samples the relevant λ-range uniformly. This allowed us to generate sufficient statistics of rare breakdown events even in the most stable regions of parameter space (see “Materials and methods” section).

The initial simulations also showed that, in the regime of significant NN repression (small κ), the expression domains of gene A at the boundaries of the system (green expression domains in Fig 2) are particularly prone to destruction by their opponent domains, as we hypothesized for two reasons: Firstly, they can expand only in one direction, towards the interior of the system; thus, unlike all the other domains, they cannot compensate shrinkage of the domain at one interface by an expansion at the other interface, and therefore take less effort to get completely destroyed. Secondly, their NN domains (B and D) are not counteracted by any strongly repressing partner at their interfaces with the A domain, which makes it easier for them to invade the A domain. We therefore decided to study a setup in which we “pin” (keep constant) the level of A proteins at the system boundaries by locally disallowing repression in the first and last rings of reaction volumes along the z-axis. This setup is not merely an ad hoc modeling assumption but motivated by experimental findings in the Drosophila gap gene system, which arguably is the most widely studied example of the alternating cushions arrangement. There, the anterior stripe of the gap gene hunchback (hb), which corresponds to gene A in our system, not only is under stringent control by Bicoid (Bcd) but additionally translated from maternal mRNA localized towards the anterior pole [65, 66]; conversely, in the posterior, zygotic hb expression is driven by a second enhancer under the control of Tailless (Tll) [67], which in turn is directly controlled by the maternal terminal system and thus tightly localized [68, 69]. Our “pinning” prescription mimics this biological situation. To assess how the assumed pinning influences our results, we later compare to simulations in which expression of A can be repressed at the system boundaries, finding our main results hold up also in this less restricted system. In particular, pinning is not necessary for enhanced stability but can further increase the maximal stability time by more than an order of magnitude compared to the system without pinning, as we present further below.

Long-term pattern stability requires optimal repression strengths

In order to see how varied repression strength affects pattern stability, we reweighted histograms of simulated trajectories over the reduced phase space spanned by order parameters λ_AC and λ_BD at different times, for different values of κ ranging from strong NN repression (κ ≃ 3) to the limit of non-interacting nearest neighbors (κ = ∞), see Fig 3. We found that there exists a region of stable expression patterns in phase space which is populated rapidly and then remains quasi-stationary, indicating that the system can remain in a metastable state if NN repression is moderate. In particular, the velocity with which the system escapes from the quasi-stationary region strongly depends on κ, with low and very high κ resulting in quick pattern deterioration, and intermediate κ values resulting in the most long-lived quasi-stationary states.

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Fig 3. Pattern breakdown in the phase space spanned by asymmetry factors.

Probability density snapshots of the phase space spanned by asymmetry factors λ_AC and λ_BC, defined in (2), at different times t for varied repression strength ratio κ. The conditions are the following: strong NN repression, κ = 3.16 (top row), optimal NN repression for pattern stability, κ = 31.6 (middle row), and lack of NN repression, κ = ∞ (bottom row). The simulation was started with the initial rectangular five-stripe A-B-C-D-A pattern (λ_AC, λ_BD) = (0.4, 0.2) (white circle) in the pinned system. All snapshots are normalized histograms of reweighted (λ_AC, λ_BD)-points within t ± 5 min. In the middle and bottom rows we identify three densely populated regions: a broad region centered around (0.30, 0.30), R_S, which contains five-stripe patterns, and two smaller regions close to (0.55, 0.30), , and (0.30, 0.55), , representing patterns with one expression domain lost (region boundaries (dashed white), details in “Materials and methods” section). Ultimately, trajectories converge towards the region centered around (0.55, 0.55), R^‡, where two domains are lost.

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Stochastic fluctuations can lead to two different events corresponding to partial pattern destruction: one in which either the A or C domain is lost first and one in which either the B or D domain is lost first. Motivated by these observations we defined a region of stable patterns in terms of the asymmetry factors as R_S ≡ {(λ_AC, λ_BD)|λ_AC ≤ 0.45 and λ_BD ≤ 0.43}, see Fig 3. States that lie outside of R_S are considered deteriorated patterns, and accordingly we also defined two regions and and a region R^‡ accumulating patterns with one expression domain lost and patterns with two domains lost, respectively. The pattern survival probability is the probability for the system to remain in the region of stable patterns until time t. We have never observed re-entry into R_S. We found that S(t) is well-described by an exponential decay, , for times t larger than a certain lag-time t_lag. k_D then defines a deterioration rate, corresponding to average pattern stability time or the mean time until the pattern has lost one of its domains, τ_D ≡ 1/k_D (see “Materials and methods” section).

By quantifying pattern stability time, we found that τ_D depends strongly on the repression strength ratio, with a maximum of τ_D as a function of κ at κ_opt ≃ 30, see Fig 4 (blue curve). For κ values close to κ_opt pattern stability is still markedly enhanced. While significantly less stable than in the region around the optimum, patterns with stability time on the order of several hours remain possible in the absence of NN repression (κ → ∞). In contrast, when NN and NNN repression have close to equal strength (κ → 1) patterns collapse almost immediately.

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Fig 4. An optimal strength of nearest neighbor repression maximizes pattern stability.

The mean time until pattern destruction τ_D as a function of κ, the ratio between the weak and strong repressor off-rate, for the system in which expression of gene A is fixed at the boundaries. We observe a pronounced maximum of the stability time when the weak repression is about 30 times weaker than the strong repression (κ_opt = 31.6) in the system with pinning at the boundaries (blue line). When pinning of the pattern at the system boundaries is relaxed (red line), the maximum of stability time moves to κ = 100. The dashed horizontal lines indicate the values for the completely uncoupled systems with κ = ∞. The dashed vertical line (orange) shows the optimal repression strength ratio predicted analytically by our stability theory, κ_theor ≃ 76 (see last part of the “Results” section).

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Examples of individual trajectories leading to (partial) pattern destruction as they proceed in biased simulation time (with increasing λ) are described in Sec S1.5 of S1 Text and shown in Figs G, H and I in S1 Text, for the most stable regime, i.e. for κ = κ_opt in the system with pinning. These examples demonstrate that multiple destruction pathways are possible in which the individual domains are destroyed in different order, and that destruction of one domain of a strongly competing gene pair can (but does not need to) facilitate subsequent destruction of a domain in the other strongly competing pair.

In the maximally stable regime restoring forces reconstitute perturbed patterns

The observation of a phase space region in which system trajectories persist for long times raises the question whether this region constitutes a true metastable basin of attraction. We first addressed this question next by analyzing transient behavior of the perturbed patterns. If enhanced phase space density in certain regions of the (λ_AC, λ_BD)-space were indeed due to the presence of a metastable basin, perturbations that transiently drive the system away from the stable pattern should be counteracted by restoring forces. To test this hypothesis, we perturbed relaxed five-stripe patterns from the hypothetical basin by artificially enlarging domains in which one gap gene is dominant. Using these perturbed states as initial conditions, we then ran the spatial-stochastic simulator with higher time resolution, and checked whether the perturbed systems relax back into the presumed basin. We investigated two types of asymmetric perturbations: “C expansion”, in which the central C domain is unidirectionally expanded at the expense of the posterior A domain, and the converse “A expansion”, in which the anterior A domain is enlarged at the expense of the C domain. The perturbation experiments are described in detail in the “Materials and methods” section, and Sec S1.2 and Fig A in S1 Text.

We find that at κ = κ_opt, for both perturbations the perturbed pattern ensembles relax back to their original positions on a timescale ∼ 10 h (see supporting Fig A in S1 Text). This demonstrates that for optimal repression strength ratio an effective restoring force counteracts deviations from the five-stripe pattern for varied λ_AC. Moreover, this suggests that the probability-enriched region within R_S is a real metastable state confined by an underlying force field. In accordance, the timescale of relaxation is orders of magnitude shorter than the timescale of pattern collapse. Thus, for κ = κ_opt pattern destruction is a Markovian transition between metastable basins with transition waiting times much longer than the timescales of intra-basin dynamics. In contrast, we could not observe clear restoring behavior in the systems with very weak or no nearest neighbor interaction. Here perturbations of similar strength tend to result in almost immediate pattern destruction.

In summary, for the repression strengths ratio κ_opt ≃ 30 that maximizes stability, pattern breakdown appears to be an activated process characterized by a restoring force towards the initial state.

Statistical analysis of phase-space dynamics reveals a metastable basin

We further figured that the existence of a true metastable basin should manifest itself also in the statistics of transient dynamics in phase space. Here the local velocities in the (λ_AC, λ_BD) phase space are particularly informative: forces that drive trajectories back into basins of attraction should translate into local mean phase space velocities with a clear bias towards the bottom of the basin.

To extract the velocity field for our system we modeled the coarse-grained pattern dynamics as overdamped diffusive motion in the plane, assuming that these degrees of freedom capture the slowest time scales of the system and making a Markov approximation for the fast dynamics [70, 71]. This technique has been successfully applied in protein folding [72–75]. The corresponding model equation is (3) where is uncorrelated (2D) white noise with unit covariance. We estimated the local drift and diffusion coefficient from our reweighed simulated trajectories by averaging local displacements (see “Materials and methods” section, and Sec S1.3 in S1 Text). Furthermore, is proportional to the effective force acting at the reduced phase space point in the overdamped Langevin model. The local mean velocity field is determined by the conditional transition probabilities between states and , and thus can be extracted from our transient simulation data. The resulting average velocity field in the reduced phase space of (λ_AC, λ_BD) for the optimal repression strength ratio (κ = 31.6) is shown in Fig 5A, and for suboptimal (κ = 1000) is shown in Fig 5B.

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Fig 5. Phase space velocity fields and passage statistics reveal metastable basins.

(A, B) Average phase space velocity fields for the system with optimal (κ = κ_opt, A) and suboptimal (κ = 1000, B) repression strength ratio, in the phase space spanned by asymmetry factors λ_AC and λ_BD as defined in (2). The small subregion with concentrically inwards-pointing velocities towards which perturbed trajectories relax, corresponding to the metastable basin of five-stripe patterns, is indicated (R_MB, dashed circle). Velocity fields were obtained by averaging displacements of all trajectories that exit the local bin (see “Materials and methods” section). Two examples of trajectories relaxing after perturbations are shown (blue lines = pert. from boundary, turquoise lines = pert. from center) with their starting points (circles). The boundaries of phase space regions (thin dashed lines) are as in Fig 3. Velocity magnitude is indicated with colors. (C) The velocity field corresponding to κ = κ_opt for the alternative asymmetry factors (“differences”) δ_AC and δ_BD, as in (4). In C the metastable basin R_MB is localized around the “center point” , corresponding to an intact pattern with equal proportions of strongly competing genes. For additional clarity, the inset in the upper left corner shows this region without the relaxing trajectories. Note the almost concentric pattern of velocity vectors pointing towards the center, highlighting the presence of the metastable basin. The magnitude unit “phase space unit per hour” (PSU/h) is specific to the chosen asymmetry factors. (D, E) The landscapes of the “pseudopotential” computed from the total number of phase space trajectories registered in the respective bin of the phase space. The contour plots to the right of the 3D views show a projected view of the same landscapes. (F, G) Comparison of sections in λ_AC and λ_BD directions, respectively, at λ_⊥ = 0.28 between the optimal and suboptimal choice of the repression strength ratio κ. Here the profile is almost identical in the metastable basin R_MB, but transitions towards the destroyed pattern states face a higher barrier in the system with optimal κ = κ_opt, in both phase space directions.

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Interestingly, in Fig 5A one can identify two regions of (λ_AC, λ_BD)-space with low average velocities: one within the region of stable states R_S, the other within the region of states in which the C expression domain is lost. The region in which either B or D are lost, has no clear boundaries for optimal κ = 31.6, and only for much larger κ = 1000 a low-velocity plateau is clearly seen in this region (Fig 3B). Notably, in the lower-left corner of the R_S plateau we notice a small region in which average velocities are significantly higher and all pointing inwards. We refer to this region as R_MB, and identify it as the metastable basin of intact, relaxed five-stripe patterns. In accordance, the two shown exemplary perturbed trajectories relax into R_MB after randomly exploring the R_S plateau, and remain confined to R_MB for later times (Fig 5A). However, if the system drifts far away from R_MB, in the direction of , the trajectories are quickly absorbed into once they reach the edge of R_S characterized by high velocity components towards .

In order to further investigate the low-velocity attraction basins and the high-velocity ridges that separate these basins, we use a different representation of pattern asymmetry, defining the shifted difference coordinates (4) These coordinates measure the deviation from an intact pattern with equal proportions of the strongly competing gene pairs, corresponding to the point in the phase space of the shifted difference coordinates, in a way that retains information about which of the antagonistic genes becomes dominant. Similar latent-space projections have recently proven instrumental in analyzing the temporal dynamics of the emerging gap gene expression pattern [76]. The corresponding average velocities in (δ_AC, δ_BD)-space are shown in Fig 5C. The low-velocity basin, corresponding to R_MB, occupies the central part of the (δ_AC, δ_BD)-space in Fig 5C. Accordingly, perturbed trajectories relax towards the region enclosed by concentric velocity vectors pointing towards P₀ (the concentric vector pattern is best seen in the inset), again highlighting the presence of the metastable basin and restoring forces that tend to drive back fluctuations that perturb the intact pattern. Overall, the finding of a metastable basin is in line with the Waddington picture of canalization [40, 41], in which developmental stages are seen as successive attractors of the underlying dynamics, with the intact five-stripe pattern considered here representing such an attractor.

In Fig 5B we show the average velocity field for the case with weaker NN repression (κ = 1000). Here the velocity fields are even more plateau-like in the region corresponding to weakly asymmetric patterns, and the characteristic concentric velocity pattern indicative of the basin in the optimal case cannot be clearly discerned any more in this case. In accordance, trajectories starting from perturbed patterns do not relax back and progress towards patterns with at least one domain lost. See also Sec S1.4 and Figs C–F in S1 Text for the corresponding velocity fields in shifted difference coordinates and additional alternative projections.

In addition to the average velocity fields of the registered phase space trajectories, the signatures of the metastable basins are also visible in the local phase space density sampled over many trajectories that explored the phase space during the whole sampled time interval, . A suitable quantity for visualizing the corresponding phase space “landscape” is the negative logarithm of ; note that in an equilibrated, stationary system this quantity would be proportional to the energy (landscape) defining the stationary probability distribution of the system. Since our system is genuinely non-stationary, this relationship does not hold. Nevertheless we can consider our most stable systems transiently equilibrated in the metastable basins or origin and akin to stationary systems until they irreversibly cross the barrier towards one of the basins corresponding to destroyed patterns. Note that the depth of these destroyed pattern basins grows with the amount of simulated time after the destruction events, because then the basins continue to be explored by phase space trajectories corresponding to fluctuations of the destroyed patterns; their apparent depth therefore depends on the prescribed maximal duration of the biased simulation trajectories in the NS-FFS scheme, which is a technical simulation parameter. In contrast, the height difference between the metastable basin of intact patterns and the barrier separating it from the destroyed patterns basin is entirely determined by the biophysical parameters that set the average time scale for the stochastic destruction process. Therefore, the barrier height, as seen from the metastable basin of intact patterns, is a biophysical property that does not depend on technical choices (but obviously the maximal simulation time has to be chosen larger than the fastest barrier crossings in order to make the barrier visible).

In Fig 5D and 5E we plot the “pseudopotential landscape” defined as for optimal κ = 31.6 (D) and suboptimal κ = 1000 (E), where is a locally smoothened version of which equalizes out small local spikes in but preserves the overall structure of the resulting landscape (see “Materials and methods” for details). The small plots right of the landscape visualizations show sections through the landscapes in direction of the asymmetry factors λ_AC and λ_BD at chosen constant values of the respective orthogonal factor (see Fig 5 caption). In both cases we can clearly identify the metastable basin of undestroyed patterns and a barrier separating it from the basins of (half-) destroyed patterns. The basin corresponding to the states in which either the B or D domain is lost is less pronounced for the optimal choice of κ due to its lower accessibility, and—more importantly—separated by a higher barrier. This is best seen in a more detailed explicit comparison of the sections through the landscapes, shown in Fig 5F and 5G. The comparison clearly reveals that the barrier separating the metastable basin of intact patterns from the basin in which the C domain is lost is both higher and wider for the optimal choice of κ, overall leading to a markedly lower rate of pattern destruction.

Taken together, the analysis of both the velocity fields and the empirically sampled phase space density demonstrate that the long-time confinement of phase space trajectories close to the five-stripe pattern at optimal NN repression is due to the existence of a metastable basin which impedes progress towards losing one of the domains by restraining the system from leaving the metastable basin. With decreasing strength of NN repression the basin gradually disappears, thus enhancing the probability of pattern deterioration.

The finding that pattern stability is enhanced by the emergence of a metastable basin is further supported by the quantification of the diffusion coefficient in space, D_λ (see Eq 3 above). As discussed in detail in Sec S1.3 and Fig B in S1 Text, we observe that the average diffusion constant in the metastable basin of intact patterns R_S, , monotonically decreases with growing κ (i.e., with decreasing NN repression strength), meaning that the estimated average time for leaving the metastable basin by random, “diffusive” motion (shown in Fig B in S1 Text, panels (b) and (c)) monotonically increases with increasing κ. Pattern stabilization around the optimal κ therefore cannot be explained by a decrease of a diffusive escape rate, but rather by the emergence of restoring forces that drive deteriorating patterns back into the metastable basin.

Stability enhancement does not require pinning

To assess whether pinning of the A-domains at the system boundaries is necessary for the observed stability enhancement at intermediate NN repression, we repeated our simulations and analysis for a system without pinning. In contrast to the system with pinning, here the promoters of gene A in the nuclei at the system boundaries can be inhibited by the repressors of A. We found that also in the system without pinning, pattern stability is markedly enhanced by the presence of weak interaction partners between two strongly repressing gene domains. In Fig 4 the red curve shows the mean destruction time τ_D against the ratio of the repressor off-rates κ for the system without pinning. We again find the highest pattern stability at an optimal repression strength ratio (red curve), which is close to the optimum in the system with pinning (, blue curve), albeit with about 10 times lower overall stability times; yet, these stability times are still about an order of magnitude larger than without fine-tuning of NN interactions.

Overall this demonstrates that enhancement of pattern stability by at least one order of magnitude is possible both with and without pinning of expression at the system boundaries. However, pinning alters the proportion of destruction pathways that the collapsing patterns pursue; in particular, it prevents the destruction of the peripherical A domain, which is the dominant destruction pathway at low and optimal κ ≲ 100 in the system without pinning. We present this effect in more detail in Sec S1.6 and Figs J and K in S1 Text.

An analytical model of expression domain competition predicts optimal pattern stability

The problem of pattern stability has been recently addressed analytically in [55], where general and exact stability conditions for a pattern of two interacting domains were derived. In that work “stability” refers not only to the robustness against perturbations, but to the ability of a pattern to survive for infinitely long time. In this section, we show that these stability conditions can be successfully applied to the multi-gene system studied in this work, in order to obtain a coarse-grained prediction of the parameter values leading to pattern stabilization.

The central result reported in [55] is the description of the dynamics of a contact zone between two gene-expression domains for various levels of mutual repression between the two expressed genes. A single expression domain can form either by overcoming the “activation threshold” in the nearby undifferentiated tissue, resulting in asymptotically constant-velocity expansion, or emerge instantaneously in the entire available tissue, when expression is constitutive (active by default). For two genes in the system (and two respective domains) the scenario depends on the strength of mutual repression. If one gene cannot prevent the expression of the other gene in the bulk of its own domain, the dominating gene overtakes the system exponentially fast, expressing in the entire volume and without forming a meaningful contact zone between domains. For stronger repression, which prevents gene expression deeper in the bulk of its adversary domain, a contact zone emerges, within which both domains of active expression overlap. However, this region of overlap grows indefinitely, albeit with asymptotically constant velocity. When the interaction strength surpasses a critical value, an asymptotically finite-size contact zone is formed. In this regime one domain can still shrink and the other grow, but in a coordinated manner, preserving the width of the contact zone. Asymptotically, the contact zone drifts with a constant velocity that is determined by the system parameters. This gives rise to a “travelling” gene expression pattern. The width and velocity of the contact zone are stable against perturbations in this phase, acting as an attractor of the system dynamics. However, while the travelling pattern is well-organized into two domains, it is not stable, in the sense that in finite-size systems it survives only for a limited time, until one domain “pushes out” the other. Finally, perfectly stable patterns arise as a special case of travelling patterns with zero-velocity drift. As such, they can survive arbitrarily long.

The simulations in this work are stochastic, tracking the chemical reactions at single-molecule resolution across the set of reaction-volumes constituting the system. However, in the limit of large particle number and small reaction volumes, this type of spatially discrete and stochastic dynamics approaches the continuous and ultimately deterministic reaction-diffusion dynamics of the type considered in [55]. The existence of this deterministic limit can be also seen as the manifestation of the emergent noise-control mechanism that overtakes the system. Therefore, we compare the numerically found optimal with the theoretically predicted to assess how well the deterministic theory approximates the dynamics in the highly stochastic regime, and to explain the nature of the emergent noise-control mechanism.

To this end, we mapped the microscopic model used in our stochastic simulations onto the effective reaction-diffusion model analysed in [55] (see “Materials and methods”). We obtained the following continuous model for the expression of each gene X in {A,B,C,D}: (5) where X₂(x, t), Y₂(x, t) are the concentration profiles of the protein dimers (indicated by the subscript 2), D is the diffusion constant, γ the degradation constant, H the production constant, and ϵ_XY are gene-gene interaction strengths. θ(…) denotes the Heaviside step function, corresponding to steep Hill-type regulatory kinetics. Note that the derivations in [55] only apply to systems with size L ≫ λ, where is the characteristic length of gene interaction. For the systems studied here, λ ≈ 8.62 μm, which is much smaller than the system size L ≃ 340 μm, warranting application of the theory.

While the original theory in [55] describes only the contact zone involving exactly two domain boundaries, we can adapt it to the four-gene system studied here. Fig 2 shows that in the alternating cushions system there are only two types of contact zones: (i) between two strongly interacting genes (NNN domains) with the third, weakly interacting gene (NN domains), expressed in the background or (ii) between two weakly interacting genes (NN domains), with all other genes having close-to-zero expression level. Thus, we will consider stability of both contact zone types separately.

In the type-(i) contact zone, the dynamics of gene expression is described by the effective equations (6) where we approximate that the third “background gene”, has a constant expression level over the contact zone. The equilibrium value of this expression level is H/γ. K_w and K_s are the weak and strong repression constants, respectively and they satisfy (cf. Eq 1 and “Materials and methods” section): (7)

Type-(i) contact zones are established between genes A and C (with B or D in the background) as well as between B and D (with C in the background).

In the type-(ii) contact zone, the equations take the form: (8) This contact zone emerges between gene pairs (A,B), (B,C), (C,D), and (D,A).

Having defined these contact zones, in the “Materials and methods” section we adapt the more general derivation of the stability conditions from [55] to the system considered here. The main idea of this derivation is that the shape of expression profile X₂(x, t), defined by Eq (5), can be found without knowing where the domain boundaries are located. Then, the positions of domain boundaries are sought from a separate set of equations. In [55], it is shown that the boundaries asymptotically travel with the common constant velocity v, preserving the distance Δr between them. This ansatz leads to an algebraic set of equations defining v and Δr, which can be solved. Eventually, the conditions for pattern stability are equivalent to ensuring that v = 0 is the correct solution. Let us define two sets of constants, one for each type of contact zone: (9) (10) and two auxiliary variables: (11) Then, the stability conditions, as derived in the “Materials and methods” section, come down to: (12) Additionally, the width of the stable contact zone reads: (13)

In order to determine the range of parameters ensuring global stability of the pattern, we apply the stability conditions (12) separately to type-(i) contact zone (Eq (9)) and type-(ii) contact zone (Eq (10)). One can notice that in each type of contact zone the equality R_X = R_Y is automatically satisfied, due to the common choice of parameters γ, H and D for both genes, as well as the symmetry in gene interactions. Sharing the same parameters is also the main reason why stability conditions (12) are much simpler than their general counterpart reported in [55]. R_X = R_Y = R means that each type of contact zone is characterized by one variable: (14) The remaining stability condition, −1 < R < 1, applied to R_(i) and R_(ii), results in the following inequalities: (15) These conditions show that the addition of weak interactions is instrumental for increasing system stability. On the one hand, the type-(i) contact zone is stable (i.e., immobile) provided that the weak interaction strength does not exceed (H/γ)⁻¹; otherwise it would prevent the expression of strongly interacting genes in this region. On the other hand, for the type-(ii) contact zone it is necessary that , as this minimal strength of repression is required to prevent co-expression of both weakly interacting genes in the same region. In order to simultaneously stabilize both types of contact zones, one needs to negotiate between these two largely opposite goals. This trade-off can be achieved only for the most marginal value in both parameter ranges, K_w = H/γ, which highlights why in the alternating cushions architecture the weak interactions have to be fine-tuned for pattern stability. In contrast, but in line with the numerical findings, the strong interactions characterized by K_s can be arbitrarily large, K_s ≤ ∞.

The simulations in this work were performed for K_s ≃ 0.003 μm⁻³ with K_w varied to obtain different values of κ, see “Materials and methods”. Calculated from these microscopic parameters, H/γ ≃ 0.23 μm⁻³. The resulting theoretical value of κ that ensures stability is then κ_theor ≃ 76. This number is of the same order of magnitude as the optimal κ in the simulated stochastic systems, showing slightly better agreement with the no-pinning case () than with the case with pinning at the boundaries (), see Fig 4.

The emergent noise-control mechanism can be understood via the analytical model

The analytical deterministic model can be employed to obtain further insights into the mechanism of increased pattern robustness against noise in the vicinity of optimal κ. For this, we must first consider the width of type-(i) and type-(ii) contact zones in their stability regions predicted by the theory from [55]. Inserting R_(i) and R_(ii) into Eq (13) with K_w = κK_s, we obtain Δr_(i) and Δr_(ii) as functions of κ, shown in Fig 6A. Here, Δr > 0 indicates a no-expression region between the domains (a gap), while Δr < 0 means that active expression regions overlap. One can instantly notice that Δr_(i) → +∞ and Δr_(ii) → −∞ at κ = κ_theor. Tending to infinite values is an artefact of our analysis, in which we treat each contact zone as a separate region, disconnected from the others. However, this behaviour conveys an important message. At κ = κ_theor the system attempts to maximize the size of each contact zone, forming five contact zones tightly filling the entire system. In this state, any pattern perturbation distorts at least two contact zones. Since each contact zone is stable, their maximized widths are attractors for the deterministic dynamics [55], and consequently the system tends to remove the perturbation. This is the origin of increased survival time of patterns at optimal κ. This restoration behavior is qualitatively similar to the model of repulsive forces between domain boundaries (kinks), discussed in [43]. Although the analytical stability theory [55] does not rely on the concept of explicit restoration forces, these forces arise effectively, leading to the occurrence of the pseudopotential in our phase space analysis, in Fig 5. Thus, the effective restoration forces form a link between the exact stability theory [55] and the approximation of interacting kinks in [43].

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Fig 6. Analysis of theoretical contact zone widths uncovers a deterministic stabilization mechanism.

(A) Plot of theoretical contact zone widths in the approximation of separate interfaces, Δr_(i) and Δr_(ii), for type-(i) contact zones (between two NNN domains with third interacting gene expressed in the background) and type-(ii) contact zones (between NN domains), calculated from Eqs (13) and (14) with κ = K_w/K_s and K_s kept constant. In their respective regimes of stability, the widths are restored by deterministic dynamics if perturbed. Vertical lines: κ₀ ≃ 38 (red, dashed) at which Δr_(ii) changes sign; critical κ_theor ≃ 76 (black, dashed) ensuring simultaneous stability of type-(i) and type-(ii) contact zones. Horizontal line (gray, dashed): limit of Δr_(i) ≃ 31.3 μm without any weak interactions (κ → ∞). (B-E) Schematic representations of system states in various regimes of κ, predicted by the deterministic model. Solid arrows: stable contact zones (restorable width); dashed arrows: unstable contact zones (non-restorable width); inward arrowheads indicate Δr_(ii) < 0; contact zones of type-(ii) (red), and type-(i) (blue). (B) 0 < κ < κ₀: type-(ii) contact zone stable, Δr_(ii) > 0, no type-(i) contact zones, domain widths lack stabilization against fluctuations. (C) κ₀ < κ < κ_theor: type-(ii) contact zone stable, partial overlap of domains, Δr_(ii) < 0, provides minimal domain width stabilization against fluctuations, but fluctuations can shift entire contact zones. (D) κ = κ_theor: type-(i) and type-(ii) contact zones stable, maximizing their widths (Δr_(i) and Δr_(ii) tend to ±∞ in the approximation of separate contact zones). The pattern is restored after any perturbation. (E) κ > κ_theor type-(i) contact zones stable, but Δr_(i) ≪ L, fluctuations can shift entire contact zones.

https://doi.org/10.1371/journal.pcbi.1012555.g006

The existence of a rigorously sharp stability condition, K_w = H/γ, raises the question about the deterministic dynamics for suboptimal choice of κ and its influence on the stochastic system. Let us first consider the case κ < κ_theor, in which type-(ii) contact zones are stable. In this regime, the system forms a pattern of domains A-B-C-D-A, but the NNN domains are so distant from each other that strong interactions are not yet important. There are four type-(ii) contact zones in this system. For κ ≃ 0, the weak interactions are extremely repressive and Δr_(ii) → +∞. Thus, the pattern collapses. For somewhat larger κ, a finite-size gap (Δr_(ii) > 0) between NN domains emerges (see Fig 6B) and is reduced to zero width (Δr_(ii) = 0) at κ₀ ≃ 38. In this regime, the pattern can survive arbitrarily long in the absence of fluctuations, but the domain widths are not stabilized in any way. Thus, in the presence of noise, the survival time of the domain depends on its size (which grows with κ), as larger domains take longer to be destroyed. For κ > κ₀, NN domains begin to overlap, as Δr_(ii) becomes negative (see Fig 6C). This marks the first emergence of the additional stabilizing mechanism, as the deterministic dynamics will tend to restore Δr_(ii) in each contact zone if perturbed. This means that, in the presence of noise, Δr_(ii) would keep returning to its deterministic value, but fluctuations can still shift a stable contact zone as one entity. If, as a result, two contact zones meet or one is pushed to the system boundary, this causes the collapse of a domain and partial desintegration of the pattern. As κ further approaches κ_theor, the overlap becomes large enough such that NNN domains begin to interact and type-(i) contact zones are formed. These contact zones have a certain minimal width, but they are not stable, in the sense that this width would not be restored if increased. The occurrence of type-(i) contact zones imposes a barrier for the further growth of Δr_(ii) with κ (see Fig 6D). At this stage, a major enhancement of pattern stability occurs, as stable type-(ii) contact zones and type-(i) contact zones tightly fill the system. At κ = κ_theor also the type-(i) contact zones gain stability, due to the further increase of overlap between NN domains. This results in the maximal global robustness of the pattern against noise. For κ > κ_theor, the width of the now-stable type-(i) contact zones quickly decreases and eventually saturates at Δr_(i) = 31.3 μm in the limit of completely absent weak interactions (κ → ∞). In this regime, with type-(i) contact zones width Δr_(i) < L/3 and no mechanism restoring the width of type-(ii) contact zones (see Fig 6E), the pattern gradually loses stability against noise. Fluctuations can shift each type-(i) contact zone as one entity (analogous behaviour to the type-(ii) interfaces in κ₀ < κ < κ_theor regime), eventually leading to pattern destruction.

In summary, the emergent deterministic dynamics, described in [55], is crucial for stabilizing the highly stochastic system simulated in this work. The increase in survival time τ_D, towards κ_opt, illustrated in Fig 4, is directly associated with the gradual activation of deterministic stabilization mechanisms, described in the paragraphs above. General principles of pattern stabilization, outlined in [55] for two genes, apply also to the four-gene system studied here, but many-gene competition and stochasticity results in a more nuanced picture of stabilization. A more detailed investigation would require considering the full spatial variability of all expression profiles together, but the approximated effective model proves useful in predicting optimal parameters.

Details of the model

Our model is inspired by arguably the most paradigmatic developmental system in which development of distinct cell fates is determined by local protein expression patterns driven by external morphogen gradients, the early embryo of the fruit fly Drosophila melanogaster. We model the egg-shaped embryo with its cortical layer of nuclei as a cylindrical array of reaction volumes coupled by diffusion of proteins. Every volume (nucleus) contains four individual promoters for each of the genes A, B, C and D. Each promoter can be repressed by the products of the three others with different affinities; this system of four mutually inhibiting genes represents the gap gene system in the early fly embryo, formed by the four genes hb, kr, kni and gt, and comprises its essential regulatory interactions. For combined repressive interactions, we employ OR-logic, i.e. whenever one of the three repressor sites is occupied expression of the gene is completely blocked. There is no competition for repressor sites on the promoters. In the unrepressed state the promoters exhibit constitutive protein production, i.e. no external activator signal is required. This deliberately mimics a situation in which activation of the genes is not provided by external morphogen gradients but by either an omnipresent master activator or auto-activation with a low activation threshold. Consequently, our model explicitly does not include morphogen gradients. As a simplifying assumption, we treat the whole production process, i.e. transcription, elongation and translation, as one step governed by a single rate β. Proteins however can form (homo)dimers and dedimerize again [93, 94], and only in their dimeric form they act as repressors. This is to ensure that antagonistic genes form bistable pairs for sufficiently strong mutual repression. Initially, all simulations are set up in a stripe pattern similar to the experimentally observed order in the embryo posterior, i.e. A-B-C-D-A [49, 53, 54]. This implies a fixed definition of “gene neighborhood” to which we refer throughout this paper: by nearest neighbors (NN) we mean the pairs (A, B), (B, C), etc., while the pairs (A, C) and (B, D) are considered next-nearest neighbors (NNN). A key ingredient of our model is that nearest-neighbor repression is weaker than repression between next-nearest neighbor domains (see “Parameter choice” in “Materials and methods”). By default we pin the expression of A at the system boundaries, i.e. in nuclei on the two outermost rings of the cylinder the A promoter is irrepressible, and therefore constitutively produces A proteins. This is motivated by the fact that in the real Drosophila embryo the gene Hb is under strict control by the maternal morphogen Bcd throughout the anterior half [65], while in the posterior a second enhancer exposes Hb to positive regulation by the maternal terminal system [67–69]. We compare this system to a system in which there is no pinning and all nuclei are identical.

Simulations

To perform rare-event sampling of the spatially resolved system we integrate our “Gap Gene Gillespie” (GGG) simulator used in previous work [14, 59] with the NS-FFS scheme [57]. NS-FFS is used to monitor and process a progress coordinate written out by GGG at regular simulation interrupts, at which GGG trajectories are cloned and restarted in a way that sampling is enhanced in the direction of increased progress coordinate, i.e. towards pattern destruction.

Parameter choice

Data analysis

Perturbation experiments

Simulations starting from perturbed initial conditions were performed directly via the GGG simulator. First the systems were relaxed to representative states within the metastable basin for a simulated time of t_relax = 30 min. The final states of these runs then were post-modified according to the following two protocols:

1. “C expansion”: starting from mid-embryo the central C protein domain was expanded as follows: the configurations in the nuclei just posterior to mid-embryo were copied and used to overwrite configurations in the subsequent Δ rows in the axial (z-) direction of the cylinder. The original configurations were stored and for each nucleus at row z_i > N_z/2 + Δ (counting from the anterior) the configuration was overwritten by the original configuration at z_i − Δ. The posterior-most nucleus was exempted from overwriting to preserve pinning.
2. “A expansion”: here the anterior A protein domain was enlarged at the expense of the C protein domain. To this purpose we applied the same copy-paste procedure as above starting form z_i = 5, however only nuclei up to mid-embryo (z_i ≤ N_z/2) were overwritten by the original configurations at z_i − Δ.

Δ quantifies the severity of perturbation. We found that Δ < 4 results in changes to the pattern that were hard to distinguish from noise, while for Δ > 12 perturbations were large enough to induce immediate pattern destruction with high probability. We therefore limited systematic tests to perturbations with Δ ∈ {4, 8, 12}. Starting from the perturbed initial conditions simulations were continued for t_sim = 20 h and snapshots of the current configurations in all nuclei were written out with an acquisition interval of t = 10 min (simulated time). 10 samples starting from 10 different perturbed initial conditions were produced for each set of parameters.

In order to overcome the difficulties of boundary detection we quantified the motion of protein domains by tracking their center of mass (CoM) along the z-axis of the cylinder. For each considered gene G we define the CoM z_G as (19) where G_tot = [G] + 2[G₂] is the total copy number. Since our system features two A domains we calculate z_A separately for the anterior (A_ant) and the posterior (A_post) part of the embryo by restricting z-integration adequately. While the CoM remains unchanged upon symmetric changes of the domain boundaries or global copy number increase, it is well-suited to indicate relaxations from the asymmetric perturbations that we apply. To find general trends in the time-evolution of the domains CoM trajectories were averaged over the 10 samples.

Effective reaction-diffusion dynamics of dimer expression

In this section we map the fully microscopic model defined in Fig 1 onto the effective model from [55]. First, we postulate that the stochastic dynamics of gene expression studied in this paper corresponds to the following effective dynamical equations: (20) where X, Y ∈ {A, B, C, D} denotes the expressed protein species, X₁, Y₁ are the concentrations of its monomer, and X₂, Y₂ are concentrations of its dimers. The synthesis and decay of dimers is described by rates k_on and k_off. The monomers and dimers degrade with rates μ_D and μ_M, respectively. In this system, only dimers are allowed to diffuse (with diffusivity D₀) and only monomers are primarily synthesized, with a maximal production rate β and production kinetics described by function f({Y₂(x, t)}_Y≠X), which we specify later. However, we assume that the production of dimer X₂ is active by default in the absence of other dimers (Y₂ ≠ X₂), so f({0}) = 1. Since the system has a cylindrical symmetry, we will treat the axis x as distinguished and treat the system as effectively one-dimensional.

The fact that the model defined by Eq (20) involves monomers and dimers complicates its mapping onto the model in [55]. We therefore translate it into a simplified model, tracking the effective dynamics of dimers only. To this end, we will first determine the ratio between stationary concentrations and in the absence of other dimers (Y₂(x, t) = 0) and assuming system homogeneity. In this case, equations (20) turn into: (21)

Solving for and , we obtain: (22)

We will now sum both equations in (20) to obtain (23) and approximate (24)

In other words, we assume that X₁(x, t) follows strictly X₂(x, t). The advantage of this approximation is that it becomes exact in the stationary state. This procedure results in the following effective equation for X₂(x, t): (25) where the rescaled constants are: (26)

We can now specify the kinetics function. The microscopic dynamics is such that each gene X is expressed, unless it is blocked by the binding of any other dimer to its repressor site on the promoter. In the averaged-out description, we expect that a sufficiently high concentration of free repressor particles effectively shuts down the production of X. Similarly to [55], we will assume that this transition is steep, so we can choose the functional form of the regulatory Hill function in (25) to have the overall shape of Heaviside step function: (27)

Finally, we relate the effective gene interaction constants ϵ_XY to microscopic parameters by the following reasoning: In the microscopic simulations, the attachment of Y₂ to the repressor site is described by the constant and detachment by or . Assuming that the repressor production speed can be approximated by Michealis-Menten kinetics, with the repressor site acting like a “catalyst”, we know that (28) where (standing for either or ) is the concentration of repressor dimers Y₂ at which the velocity of production of Y₁ is at the half of its maximal value. We postulate that at this point Y₂ effectively switches off the production of X, and we equate this point with reaching the threshold for production in (27). Hence, the following is satisfied: (29)

Solving for ϵ_XY we obtain: (30)

We choose C_X < 0 to ensure that the production of X is active by default, in the absence of repressive dimers (Y₂(x, t) = 0). Since C_X is now present in every term in (27), we can factor it out and neglect. Taken together, and assuming that diffusion, degradation and production constants are the same for all genes, that is: D_X = D, γ_X = γ and H_X = H for all X ∈ {A, B, C, D}, the microscopic dynamics of gene expression mapped onto the effective model results in Eq (5).

Derivation of stability conditions for a contact zone between two domains

This section outlines the origin of stability conditions for the effective continuous model, derived in the previous section, which is employed for the analysis of the four-gene pattern in the approximation of separate contact zones. We discuss the major steps leading to stability conditions in the current case, taking advantage of the specific setting of the system studied in this work. For a detailed and more general derivation we refer the reader to our earlier work [55].

Let us consider a pair of genes (X,Y), whose expression dynamics is described by the effective equations: (31) These equations describe both type-(i) and type-(ii) contact zones, albeit for different values of constants (see Eqs (9) and (10)). We assume that the system is open (L → ∞) and two respective expression domains occupy the opposite ‘ends’ of the system. That is, initially, the expression profiles of the respective dimers read: (32) Here A_X and A_Y are initial amplitudes, sufficiently high to initiate auto-activation, while q_X(t) and q_Y(t) denote the positions of the expression domain boundaries.

The effective equations in the form (5) can be solved analytically in this system, providing the spatio-temporal profile of expression for both dimers: (33) where the Green’s function of Eq (5) in the open system reads: (34) However, the solution (33) is known only up to the position of the domain boundaries, q_X(t) and q_Y(t). In order to determine these positions, the “Free Boundary Problem” must be solved, defined by the activation conditions at each boundary: (35) Inserting the solution (33) into these equations leads to a system of coupled nonlinear integral equations. For t large enough such that the system ‘forgets’ its initial conditions, these equations simplify into: (36) In [55] we show that the asymptotic solution of these equations is provided by a constant velocity ansatz, and , where and are constants. For this choice, the right-hand side integrals in Eq (36) saturate at constant values, though corresponding to domain boundaries travelling with constant velocities. Moreover, for sufficiently strong interactions between the genes, these velocities must be equal, v_X = v_Y = v, meaning that the domains change their size in a coordinated manner. For the constant velocity ansatz, the integrals can be analytically computed. This turns the system of integral equations into an algebraic system, defining the common velocity v and the distance between the boundaries . In the t → ∞ limit, this system reads: (37) Finally, the patterns for which v = 0 are stable, that is, they do not change in the long-time limit. Substituting v = 0 turns Eq (37) into: (38) where R_X and R_Y are defined as in Eq (11). In order to make v = 0 the solution of Eqs (37), Eqs (38) must be solvable and satisfied by the same Δr, as the system is over-defined. This happens, provided that (39) which constitutes the stability conditions utilized in this work. Further, we can also derive: (40) However, one can notice in Eq (38) that 1 − e^−|Δr|/λ > 0 for any Δr, thus: (41) This allows us to obtain Δr, as provided by the formula (13).