Linear framework models

We model ligand-binding-readout systems as discrete-state, continuous-time Markov processes using the graph-theoretic linear framework [17–21]. This framework describes the dynamical properties of a Markov process on a graph, such as its steady-state behaviour or its first-passage times, in terms of the structural properties of the graph. Crucially, this framework allows for the derivation of exact, closed-form expressions for these dynamical properties, obviating the need for specifying numerical values for model parameters. We have extensively exploited these capabilities in previous studies to analyse various types of input-output systems, including those with multiple binding sites [5] and conformations [24,25].

We start by describing the system as a finite, directed graph, G, with labelled edges, in which the vertices, , represent the system states; the edges, denoted , represent transitions among the states; and the edge labels, denoted , represent transition rates with dimensions of . Such a description gives rise to a corresponding continuous-time Markov process, , on state space , in which the edges represent possible transitions between these states, with each edge label as the corresponding transition rate. To be more precise, the infinitesimal transition rate from i to j is nonzero if, and only if, the edge exists in G, and the rate is given by the edge label:

We note that the edge labels, , are assumed to be constant in time. As such, the transition rates in are constant, i.e., is a time-homogeneous Markov process.

Now, suppose that , and let p_i(t) be the probability that occupies vertex at time t, given some choice of initial vertex. The time-evolution of this probability is given by the master equation [26,18],

(1)

where and is the n × n Laplacian matrix of G, whose entries are given by

(2)

For instance, we can use this formula to see that the Laplacian matrix of the three-vertex graph, , in Fig 2A is the following 3 × 3 matrix,

(3)

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Fig 2. Using spanning trees and forests to calculate steady-state responses and activation times.

(A) An example three-vertex graph, , and a corresponding augmented graph, , as discussed in the text. Here, . (B) The spanning trees of rooted at 1, 2 and 3. Roots are shown in orange. These spanning trees contribute to the calculation of the steady-state response, through Eq. 9; see the text for details. (C) The spanning forests of rooted at {j,4}, for each j = 1, 2, 3, in which there is a path from vertex 1 to vertex j (top three rows); and the spanning trees of rooted at 4 (bottom row). Roots are shown in orange. These spanning forests contribute to the calculation of the activation time, through Eq. 10; see the text for details.

https://doi.org/10.1371/journal.pcbi.1014288.g002

We will consider two kinds of models in our analysis. First, we consider chain models (also called pipeline models in previous work [20,21]), denoted , in which N vertices are reversibly connected in a linear arrangement. In particular, the vertices of are given by , and consecutive vertices are connected by reversible edges, and for (Fig 1D–E). Each vertex is assumed to correspond to a different state of the system, including the presence or absence of bound ligand, in addition to other internal features. For example, in a receptor system, a state could correspond to a given conformation or post-translational modification state. In a gene regulation system, states could correspond to DNA conformations, nucleosome patterns, polymerase occupancy at the promoter, and other salient features. The binding of the ligand is modelled through the edge labels, which can be generic functions of the ligand concentration, (Fig 1D–E). For the sake of mechanistic generality, we leave the functional form of unspecified.

In order to formalise a particular assumption for how the ligand influences the system, we turn to ladder models, denoted , which explicitly incorporate ligand binding to a single site, and assume that the ligand has an effect while bound (Fig 1F–G) [13,27]. In particular, is a graph on 2N vertices,

where represent ligand-unbound states and represent ligand-bound states. There are edges between pairs of consecutive unbound vertices, and ; pairs of consecutive bound vertices, and ; and pairs of unbound and bound vertices of the same index, and . We assume that the label on each binding edge, , and unbinding edge, , is independent of the index i, as

where x is the ligand concentration, k_on is a rate constant with units of , and k_off is an off-rate. On the other hand, we allow for the other edge labels to vary with i, and write them as

Here, is a dimensionless parameter that captures the extent to which the ligand promotes (), hinders (), or maintains as is () the transition from index i to index j. We call these parameters regulatory factors.

Quantifying steady-state level and activation time

We now describe how we quantify the steady-state response and activation time of these systems. Each system we consider in this paper produces a molecular readout, M, whose copy-number we denote by n_M. We assume that each graph, G, contains a subset of vertices, , from which the system can produce M at a constant rate r. In addition, we assume that M undergoes first-order degradation, with rate . In this context, we can naturally define the steady-state level, SS(x), of M as the steady-state mean value of n_M, which can be shown to be equal to (Appendix A in S1 Text) [28,29],

(4)

where is the steady-state probability of the vertex v. That is, is the v-th entry in the vector, , of probabilities that satisfies Eq. 1 with the left-hand time-derivative set to zero,

(5)

Eq. 4 describes how the steady-state behaviour of the input-output system, as described by the graph G, determines the steady-state level of the readout, M, as a function of the input concentration, x. For the chain models, we assume that ; for the ladder models, we assume that .

To obtain for each , Eq. 5 tells us that p^* lies in the kernel of . If G is strongly connected—that is, if every pair of vertices in G is connected by a path of directed edges—then one can show that this kernel has dimension 1 [18],

In this case, p^* is unique, and can be obtained by identifying any vector, , in by normalizing by the coordinate sum. The kinds of graphs we consider, and , are strongly connected for all N. To calculate in the numerical analyses we perform below, we obtained the singular value decomposition (SVD) of and set to the right singular vector corresponding to the zero singular value [30] (Materials and Methods).

On the other hand, we define the activation time of a system to produce one new molecule of M as a mean first-passage time (mFPT) on an augmented graph, G⁺, in which an additional “terminal” vertex is introduced to describe the production event. We also denote this new vertex by M, and identify it with n + 1 in the vertex ordering. We then define G⁺ as the graph on vertices that is obtained by adding the edges , for , each with the production rate, r, as the edge label. Fig 2A demonstrates this construction on our example graph, , with . The resulting augmented graph, , contains a new terminal vertex, M = 4, as well as the edge with label . From Eq. 2, we can see that the Laplacian matrix of is the 4 × 4 matrix,

(6)

Now, let be a non-terminal vertex in G⁺, and let be the Markov process associated with G⁺. We define the activation time from vertex i as the mean time taken by to first reach M from i. In other words, the activation time from vertex i is the mFPT,

(7)

This mFPT can be obtained from the matrix equation (Appendix A in S1 Text) [21],

(8)

where we have introduced , and is the n × n sub-matrix of L(G⁺) obtained by removing the row and column corresponding to . Briefly, this matrix equation is obtained by defining the adjoint master equation of and taking its Laplace transform (Appendix A in S1 Text). For our numerical analyses below, we solved for the vector of mFPTs in Eq. 8 by obtaining the QR decomposition of the left-hand matrix [30] (Materials and Methods).

Let us consider how this calculation bears out for our example graph, . Taking the negative transpose of the Laplacian matrix, , in Eq. 6 and removing the row and column corresponding to M = 4, we get

As such, the vector of mFPTs to M from the other three vertices satisfies the equation (Eq. 8),

Analytical formulas for SS(x) and mFPTⁱ(x)

As described above, it is straightforward to solve Eqs. 5 and 8 numerically, given particular values of the edge labels, using standard techniques. Complementing this approach, we also sought to obtain analytical formulas for SS(x) and mFPTⁱ(x), both of which are accessible via the Matrix-Tree theorems [18,21]. To understand these formulas, we must first consider the spanning forests of a graph. For any graph , a spanning forest of is a subgraph, F, that (1) contains all the vertices in (“spanning”); (2) contains no cycles, even when ignoring edge directions (“forest”); and (3) contains exactly one outgoing edge from each vertex other than a subset, , from which there are no outgoing edges. This subset of vertices are called the roots of F. If consists of a single vertex, then F is a spanning tree. We denote the set of spanning forests of rooted at by . Returning to our running example, we show in Fig 2B the set of spanning trees of that are rooted at each of the vertices 1, 2 and 3, which we denote by , and , respectively.

Since there is exactly one outgoing edge from each non-root vertex in any spanning forest F, it is easy to see that F must contain a directed path of edges, , from each vertex, , to precisely one root, . For instance, if F is a spanning tree rooted at , then every vertex must have a path to j in F. On the other hand, each root in any spanning forest has a trivial path to itself and, since roots lack outgoing edges, evidently has no path to any other root. Now, given and , we denote by the subset of spanning forests rooted at A in which there is a path from i to j. Here, i may either be any non-root vertex or j itself, so that . If i = j, then, as described above, j always has a trivial path to itself and has no path to any other root, and so we have .

Fig 2C shows four subsets of spanning forests of our example augmented graph, , each with a different set of roots (, , , and ) and a different choice of root to which there is a path from the vertex 1. For instance, the second row contains the spanning forests of that are rooted at and contain a path from 1 to 2; following our notation, this is the set . Note that, in the first row, we show the set of spanning forests rooted at with a path from 1 to itself; as described above, since such a trivial path exists in every spanning forest rooted at , this set is equal to the set . Similarly, the last row shows the set of spanning trees rooted at 4; since every vertex has a path to the lone root in a spanning tree, this set is also equal to .

We are now ready to provide our formulas for SS(x) and mFPTⁱ(x). For the former, the Matrix-Tree theorem tells us that the steady-state probability vector, p^*, in Eq. 5 is given by [17–19]

(9)

where is the weight function, which, here, evaluates the sum of products of edge labels over each spanning forest in . More broadly, if is any collection of graphs, then is defined as

To see how this calculation works on , we can run through the spanning trees in Fig 2B and read off the corresponding edge labels in Fig 2A, to get

We can then use Eq. 9 to calculate p^*, then use Eq. 4 to solve for SS(x). In particular, since , Eq. 4 tells us that

As for the mFPTs, we can use the All-Minors Matrix-Tree theorem to show that [21]

(10)

where, as described above, G⁺ is obtained by augmenting G with the terminal vertex, M (Appendix A in S1 Text). For instance, applying this formula to our example augmented graph, , with i = 1 yields

(11)

The spanning forests of that contribute to the numerator are given in the first three rows of Fig 2C; the spanning trees that contribute to the denominator are given in the last row. Reading off the edge labels in Fig 2A, we can see that these terms are given by

which we can substitute into Eq. 11 to obtain an expression for mFPT¹.

Eqs. 9 and 10 demonstrate that analytical formulas for both quantities, in terms of the edge labels of the underlying graph, can be obtained through spanning tree/forest enumeration, as we have done with and . However, such enumeration can be prohibitively expensive even for simple graphs. To circumvent this, we turn to a recurrence relation due to Chebotarev and Agaev [31], which we summarize here and describe in more detail in Appendix A in S1 Text.

For any graph with vertices , Chebotarev and Agaev defined a sequence of n × n matrices, , in which the (i,j)-th entry, denoted by , is the total weight of the spanning forests of with the following properties: (1) it contains k edges, (2) j is one of its roots, and (3) it contains a path from i to j. Upon evaluating these matrices for G and G⁺, we can rewrite Eqs. 9 and 10 in terms of these matrices (Appendix A in S1 Text). First, we can rewrite our expression for in Eq. 9 as

(12)

Second, we can rewrite Eq. 10 as

(13)

Now, these reformulations are useful because Chebotarev and Agaev showed that these matrices follow the recurrence relation,

(14)

with the initial condition . This means that we can calculate , , and by iteratively applying Eq. 14, then use their entries to calculate in Eq. 12 and therefore the steady-state response (Eqs. 4 and 9), as well as the mFPT with Eq. 13. Crucially, these equations constitute an algorithm with a polynomial runtime in the number of vertices, n, that circumvents the need to enumerate spanning trees and forests.

Let us now consider how this calculation proceeds on and . First, taking the negative transpose of in Eq. 3 and in Eq. 6, we get

and

Noting that , we can now apply Eq. 14 to compute , and . The resulting matrices are rather complicated, but the reader can easily undertake these calculations to verify that the entries required in Eqs. 12 and 13 are given by

The reader may also verify that these expressions are precisely the weights of the spanning forests given in Fig 2B and C. S1 Fig shows the agreement between the mFPT obtained using this procedure and a set of simulated trajectories of using the Gillespie algorithm [32].

Quantifying decoupling

To quantify decoupling between SS(x) and mFPTⁱ(x), we consider normalised dynamic ranges for the two outputs, which we define as

(15)

These definitions restrict and to between 0 and 1. Indeed, this restriction is self-evident for ; for , this stems from Eq. 4, which tells us that

is the sum of steady-state probabilities for the subset of productive states, . From here, we define the dynamic range of the two outputs as

(16)

which also range between 0 and 1. We note that both and are relative dynamic ranges: the former is normalized by the theoretical maximum value of SS(x) (namely ), while the latter is normalized by the actual maximum value of mFPTⁱ(x). Therefore, both dynamic ranges account for changes in SS(x) and mFPTⁱ(x) in proportion to these maximum values. We also note that perfect decoupling, in which only the steady-state level changes with input concentration, is obtained when and .

Decoupling under rate scale separation in and

We start our analysis by considering the two-state chain model, , which is equivalent to the “random telegraph” model of transcriptional bursting (Fig 1E) [14,33,34]. Here, vertex 1 represents an inactive state, and vertex 2 represents an active state that can produce the molecular readout, M, so that . The steady-state level is given by (Eqs. 4 and 9; Appendix A in S1 Text) [22,28]:

whereas the activation time is given by (Eq. 10)

(17)

where we have set the initial vertex to i = 1. Notice that, upon sending , we have

(18)

From this, we can see that if the ligand acts only on the edge , such that does not depend on x, and the production rate is large (), then mFPT¹(x) remains constant while the steady-state level, SS(x), changes with x. Thus, in this regime, the ligand may only affect the steady-state level, but not the activation time, leading to decoupling. This simple model immediately suggests that decoupling can be easily achieved if the ligand only acts on the deactivation transition, as long as production from vertex 2 is sufficiently fast such that, upon reaching vertex 2, the system produces M before transitioning back to vertex 1. We call this rate scale separation. If the ligand acts on the edge as well, so that also depends on x, whether decoupling is possible depends on the functional forms of and .

In order to examine the implications of assuming a specific mechanism by which the bound ligand affects the system, we next considered the corresponding ladder model, (Fig 1F). Here, recalling that , the steady-state level of M is defined as (Eqs. 4 and 9)

and we can similarly use Eq. 10 to derive the activation time, , with initial vertex i = U₁.

The ligand can either promote or hinder production of the readout; in this model, this is determined by the values of the regulatory factors, and . The former measures the strength with which the ligand regulates the forward transition, ; the latter measures the strength with which the ligand regulates the backward transition, . To simplify our analysis, we will assume that the ligand acts on only one of the two transitions, and that it promotes readout production, thus acting as an activator. Mathematically, this means that either and , or and . Using the mathematical machinery laid out in the previous section, it can be shown that, in this regime, SS(x) increases monotonically with x and decreases monotonically with x (Appendix B in S1 Text). In this case, analytical expressions for the corresponding dynamic ranges can be obtained by comparing the values of and at x = 0 and . In particular, when the forward transition is regulated by the ligand (, ), we obtain the following dynamic ranges:

(19)

Meanwhile, when the ligand regulates the backward transition (, ), we obtain,

(20)

The expressions in Eq. 20 reveal that, when the backward transition is regulated by the ligand, tends to zero as , while does not depend on r. However, if the forward transition is regulated by the ligand (Eq. 19), then taking the same limit causes to converge to a finite nonzero value, namely . Therefore, a similar result to what we found for holds for : the two outputs can be decoupled if the ligand regulates the backward transition, , and the production rate, r, is large. Notice that, in both equations, and may both tend to zero in other limiting regimes, e.g., ; this does not correspond to our definition of decoupling, but rather a trivial case of unresponsiveness of the system.

Decoupling under rate scale separation in

Molecular systems in various biological settings often transition through multiple states before producing the molecular readout [9,14,15,35,36]. We thus asked what happens in the ladder model .

To begin with a simplified setting, we assumed and , and that the ligand regulates exactly one of the four transitions, , , , and , promoting readout production. In this case, there are four possibilities, which are enumerated in Table 1. For each of these regimes, it can be shown that SS(x) and are both monotonic in x (Appendix B in S1 Text) and analytical expressions for the dynamic ranges of the corresponding normalised quantities can be obtained, analogously to the model.

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Table 1. Dynamic ranges of and in , for regulatory regimes that promote readout production through the regulation of one transition.

https://doi.org/10.1371/journal.pcbi.1014288.t001

The normalised dynamic ranges for the four parametric regimes are summarized in Table 1 below, where we have introduced the dimensionless parameters,

(21)

We first notice that, in almost all the cases, having a large production rate () is no longer sufficient to get decoupling. The only case in which this is sufficient is case 1.IV, where only the last backward transition, , is regulated. Here, we may rewrite as,

This is in line with the findings for in the previous section, and can be understood intuitively as follows. When r is sufficiently large, every time the system reaches U₃ or B₃, it will rapidly proceed to M without backtracking to vertex U₂ or B₂, respectively. Therefore, the mFPT from U₁ to M can be approximated as the mFPT from U₁ to U₃ or B₃, whichever is reached first. Now, if the ligand does not regulate any transition other than , the dynamics with which the system proceeds to U₃ or B₃ in the limits of zero or infinite ligand concentration, respectively, are the same on average. Since the mFPT is monotonic in x (Appendix B in S1 Text), this implies that the mFPT does not change with x. Therefore, in case 1.IV, a large production rate is sufficient for decoupling.

In addition, we notice that for cases 1.II, 1.III and 1.IV, but not 1.I, there exists a different parametric regime in which tends to zero but does not, thus giving rise to decoupling: , which we may rewrite in terms of rates as . In particular, we note that, in cases 1.II and 1.III, it is not sufficient to have either or ; rather, both and must be large. In this rate-scale-separated regime, the first forward transition in the absence of ligand, , is much slower than the second, , as well as the production transitions, and . In addition, when (as in cases 1.II, 1.III, and 1.IV), the first forward transition in the presence of ligand, , is also much slower than , , , and . We hypothesised that, in this case, a partitioning of the graph arises where the slow rate, namely , completely determines the activation time, while the fast rates dictate the dynamic range of the steady-state level. To test this hypothesis, we examined for cases 1.II, 1.III and 1.IV, when x = 0. For all three cases, the activation time is given by (Eq. 13 and setting x = 0),

(22)

Here, imposing yields an activation time that depends merely on . Since, as shown above, in this parametric regime for cases 1.II, 1.III and 1.IV, this means that depends merely on for all x. Therefore, under rate scale separation, the activation time depends entirely on the slower, unregulated forward transition.

Meanwhile, the expression for the steady-state level, SS(x), depends on the specific case, as well as which parametric limits are applied to realise the condition that and should be large. For instance, in the simple setting in which we send , it can be easily shown that the steady-state level in case 1.III—obtained by applying Eq. 9 and taking the symbolic limit as —depends on the fast rate , and the corresponding regulatory factor, , in addition to k_on, k_off, and x. In contrast, the steady-state level in case 1.II, in which the slow backward transition is regulated, depends on the regulatory factor, , but not the fast rate, .

Next, we aimed to assess whether rate scale separation can still give rise to decoupling if we relax some of the assumptions made in the above analysis, namely that and . We found that, upon relaxing these assumptions, the expressions for and become substantially more complicated; therefore, we resorted to numerical optimization to search for parameter regimes that lead to decoupling, focusing on case 1.III (; Fig 3A).

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Fig 3. Decoupling under rate scale separation in the ladder model, , for case 1.III in Table 1 (A–E) and a generalization of case 2.I in Table 2 (F–J).

(A) Schematic of with regulation of (case 1.III). (B) Distribution of coupling scores after termination of the PSO. Each PSO run was terminated whenever f < 1 for more than 5 consecutive generations. For almost all the runs (n = 99) f < 0.1. (C) Evolution of f over each PSO run. The red curve represents the “best” optimized parameter set with the smallest value of f. (D) Input-output responses of optimized parameter sets for which f < 1 and the steady-state level increases monotonically with x. (E) Distributions of parameter values corresponding to the curves in panel D, with and . The green curve represents the best parameter set (red curve in C). (F) Schematic of with regulation of and (generalization of case 2.I). (G) Distribution of coupling scores after termination of the PSO. Each PSO run was terminated after 23 hours of computation time. For almost all the runs (n = 99) f < 0.1. (H) Evolution of f over each PSO run. The red curve represents the best parameter set. (I) Distributions of optimized parameter values for which f < 1 and the steady-state level increases monotonically with x. The green curve corresponds to the best parameter set (red curve in H). (J) Normalized dynamic ranges for two families of parameter sets, with the parameters set as follows: , , , , , or 10, and varied over a logarithmic range. The dots represent numerical computations (Materials and Methods), and the dashed lines represent the formulas in Table 2 (case 2.I).

https://doi.org/10.1371/journal.pcbi.1014288.g003

We defined a coupling score,

which describes the extent of coupling between the steady-state level and activation time. Perfect decoupling corresponds to f = 0, corresponding to a maximum steady-state dynamic range () and a minimum activation time dynamic range (). We then searched for parameter sets that yield a small value of f by using a Particle Swarm Optimization (PSO) algorithm [37,38] (Materials and Methods). Briefly, this algorithm begins with a collection (or “swarm”) of parameter sets (or “particles”), and iteratively updates each particle’s position and velocity according to the values of the objective function (here, f) across the swarm. Over successive generations, the swarm collectively converges to the optimal solution(s) with respect to the objective function.

We allowed the rates, , k_off, and r, to lie within a large parameter range, namely in units of . The binding rate constant, k_on, was also assumed to lie in the range , but in units of , where “c.u.” denotes the concentration units used for x. Finally, the dimensionless regulatory parameter, , was constrained to lie in the range . Each parameter was restricted to lie within these ranges throughout the optimization. We ran 100 independent optimization runs, each starting from a different random initial condition. Fig 3B and C show that more than 90% of the runs converged to a final coupling score of f < 0.1, strongly suggesting that this procedure effectively minimizes the objective function.

Among these successful optimization runs, we retained only the optimised parameter sets for which increases with x, and visualised the corresponding normalised steady-state level and activation time responses (Fig 3D). This revealed that, while the steady-state level increases monotonically with x (Fig 3D, left), the mFPT barely changes (Fig 3D, right), demonstrating near-perfect decoupling. In line with our analytical results, we found that (1) the values of and are both large, and (2) the second forward transition, , is strongly promoted by the ligand, with almost always reaching its maximum value of 10³ (Fig 3E). This strongly suggests that our optimization procedure is identifying parameter regimes that achieve decoupling via rate scale separation.

Notably, we also found that often exceeds , albeit to an extent less than r; for instance, the best parameter set over all optimisation runs (Fig 3E, green) exhibited and . We also found a broad distribution of values for . This indicates that the assumptions that and are not necessary for decoupling via rate scale separation. Indeed, combining these results with the analytical formulas in Table 1 reveals that strengthens decoupling, by increasing the steady-state dynamic range, . We can understand this effect by considering some simple limiting cases. For instance, it is easy to see that, for a regulatory regime in case 1.III in which , , and , we have a basal steady-state response of , which limits the steady-state dynamic range that can be achieved (Appendix C in S1 Text). On the other hand, removing the equality constraints on the horizontal transition rates enables the accumulation of steady-state probability on the non-productive states, U₁ and U₂, and thereby decreases , which allows for an increased dynamic range. Therefore, asymmetry in the transition rates acts in concert with the regulatory factor, , to modulate the steady-state dynamic range, whereas and together dictate the mFPT dynamic range, thus inducing decoupling.

We next proceeded to extend our analysis to regulatory regimes in which the ligand regulates two transitions. There are 16 such possible regulatory regimes, depending on the choice of ligand-regulated transitions and whether each corresponding regulatory factor, , is either greater than or less than 1. We can categorize these regimes into two classes: eight coherent regimes, in which the ligand consistently promotes or hinders transitioning towards the productive state (e.g., and ); and eight incoherent regimes, in which the ligand simultaneously promotes and hinders transitioning towards the productive state (e.g., and ) [24]. We first restricted our attention to the four coherent regulatory regimes in which the ligand consistently promotes transitioning towards the productive state; these regimes are enumerated in Table 2. For each of these regimes, it can be shown that SS(x) and increase and decrease monotonically with x, respectively (Appendix B in S1 Text), and we can obtain analytical expressions for the normalised dynamic ranges of these quantities, as before. These expressions are given in Table 2.

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Table 2. Dynamic ranges of and in , for regulatory regimes that promote readout production through the regulation of two transitions.

https://doi.org/10.1371/journal.pcbi.1014288.t002

Although these expressions are more complicated, we still see that imposing rate scale separation, by setting , causes , but not , to tend to zero, as long as the slow, forward transition () is not regulated (cases 2.III and 2.IV). Moreover, for each of these two cases, it is easy to directly compare the expressions for and to the expressions that arise when only one transition is regulated (cases 1.II and 1.III for 2.III, and cases 1.II and 1.IV for 2.IV), to see that decreases and increases when a second transition is regulated. As such, introducing a second regulated transition enhances decoupling.

We next assessed the relevance of the constraints and , again using the numerical optimization procedure outlined above. We aimed to minimize the coupling score, f, for a generalized version of case 2.I (Fig 3F), in which and can both assume any value within the range, . Similarly to the optimization for case 1.III, we found that more than 90% of the optimization runs converge to a coupling score of f < 0.1 (Fig 3G), albeit with a larger number of generations (Fig 3H), as expected by the increased dimensionality of the parameter space. Notably, we found that the optimal parameter sets exhibit significant rate scale separation, and , as well as values of , representing little to no regulation of the slow forward transition , and close to the maximum value of 10³ (Fig 3I). This strongly suggests that, to attain decoupling, the algorithm is effectively reducing this generalization of case 2.I to a regulatory regime in case 1.III, in which is unregulated. Therefore, regulation of the slow forward transition, , does not improve decoupling when the input also regulates . This contrasts with what we observed above for cases 2.III and 2.IV, where regulation of two transitions does improve decoupling when is not regulated.

In addition, we found that the ratios and follow similar distributions as in case 1.III (Fig 3E), with in the best parameter set (Fig 3I, green). This, again, reflects the fact that increasing decreases , and therefore increases , thus strengthening decoupling.

The equations in Table 1 and Table 2 and the numerical results in Fig 3 show that a large rate scale separation can give rise to decoupling. To assess whether decoupling can be achieved in a more constrained scenario, we examined a family of example parameter sets in case 1.III with , , , and , and plotted and while varying over several orders of magnitude (Fig 3J, dark colours). Note that, in this case, . Here, we found that, while did not significantly vary with , we could decrease to as small as 0.1 by setting , suggesting that values of , , and much less than those reported in Fig 3E can also give rise to significant decoupling. Meanwhile, we found that the same family of parameter sets but with , which instead fall under case 2.I, did not exhibit significant decoupling for any choice of (Fig 3J, light colours), consistent with our observations in Fig 3I.

In summary, our numerical and analytical results demonstrate that rate scale separation, when paired with a lack of regulation of the slower forward transition, , gives rise to decoupling in the ladder model, . In order to check whether this same mechanism enables decoupling in larger models, we performed a similar analysis of and found that, there too, decoupling can arise when (1) the system exhibits a block of transitions that are slower than a subsequent block of transitions, and (2) regulation occurs along one or more of the faster forward transitions (S9A–C Fig). Such separations of timescales have been widely recognised in the setting of gene regulation, where changes in chromatin state are typically measured to proceed on slower timescales than TF binding and the reactions that comprise the polymerase cycle [33,39–41]. This suggests that decoupling in gene regulation could arise when activating TFs do not accelerate the slow chromatin opening transitions, but rather work on other subsequent steps.

Decoupling due to incoherent regulation, in the absence of rate scale separation

We then asked whether there are alternative regulatory mechanisms that can yield decoupling when all transitions operate on similar timescales. Recently, we and others have argued that TFs may act on multiple steps of a gene-regulatory mechanism in an incoherent fashion, simultaneously promoting and hindering transcription [22,24]. In the light of this, we hypothesized that such incoherent regulation may be an alternative way to maintain a constant activation time, perhaps by counterbalancing the effects of promoting progression towards the productive state through certain transitions by hindering this progression along other transitions, all while allowing for a change in the steady state.

To examine how decoupling might arise when the transition rates are constrained to be similar, we first considered the extreme scenario in which . In this case, assuming that the ligand acts only on the forward transitions and (i.e., and ), it can be shown that decoupling cannot be achieved in any coherent regulatory regime in which and . In particular, it can be shown that, in this regime (Appendix D in S1 Text),

This implies that, whenever and , the coupling score, f, must be greater than one. Therefore, if the ligand is assumed to promote one of the forward transitions and , decoupling may only be achieved if the ligand hinders the other forward transition, i.e., the ligand regulates the two transitions in an incoherent manner.

We next considered what happens when we allow for such incoherent regulation by the ligand. Here, the ligand regulates two transitions in such a way that it promotes progression towards the productive state through one transition, while hindering this progression through the other (e.g., and ). In this case, and are not necessarily monotonic in x (Appendix B in S1 Text), consistent with similar observations reported in previous work [23,24]. However, we can still characterise conditions under which is not necessarily constant in x, but satisfies the weaker condition,

(23)

In particular, we can show that, if we set and allow for regulation along and ( and ), then we have,

Equating this to zero and solving for either or , we obtain,

From here, it is easy to see that if, and only if, . This implies that any coherent regulatory regime in which or cannot satisfy Eq. 23, and therefore cannot exhibit a constant in x. This demonstrates that, in the extreme scenario where , incoherent regulation is necessary to achieve , which is a prerequisite for a flat activation time, . We emphasize, however, that may be non-monotonic in x in general, in which case may be nonzero even if the initial and final values of are the same.

To pursue a more comprehensive analysis of decoupling in the situation where the transitions proceed with similar rates, we again turned to numerical optimisation. In particular, we adapted our PSO approach to incorporate a “Rate Scale Constraint” (RSC) that constrains the ratio between each pair of transition rates, as follows:

(24)

where RSC is some positive constant. The smaller RSC is, the more similar , , , and tend to be. Within the PyMoo optimisation framework that we used to perform PSO [38], inequality constraints are handled as penalties to the objective function; as such, we independently confirmed that all solutions obtained from the PSO do satisfy the constraints given in Eq. 24 (S2C Fig, left). We emphasise that we did not impose any constraints on the ligand’s regulatory mode, in principle allowing for both coherent and incoherent regulation.

We first focused on the case where the ligand may regulate the two forward transitions, and (so that and/or may be distinct from 1, and ). We ran PSO with six different values for RSC: 0.005, 0.05, 0.5, 1, 2, and 3. As before, to increase the probability of finding global optima, we performed 100 replicates of the optimization, each starting from a different random initial condition. The convergence of each replicate for RSC = 0.005 is shown in S2A Fig. Consistent with the analyses throughout the manuscript, we considered only activating responses, where the steady-state response increases with TF concentration.

Fig 4A shows that, as , we obtained a larger minimum coupling score, suggesting that it is more difficult to obtain decoupling under rate scale constraint. When examining the corresponding input-output curves for RSC = 0.005 in which increases with x, we observed some dependence of the normalised mFPT on the ligand concentration (Fig 4B, bottom). Yet, we observed that the increase in the coupling score was mostly determined by a smaller dynamic range in the steady-state level, arising from a nonzero basal steady-state level at zero input concentration (Fig 4B, top). This is consistent with our previous reasoning: the closer and are to 1, the closer the normalised steady-state level at zero ligand concentration is to 1/3, which is indeed the value of that we observe in our optimisation results (Fig 4B, top).

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Fig 4. Decoupling under rate scale constraint arises under incoherent regulation.

(A) Distributions of the coupling score, f, obtained from optimization with various rate scale constraints (RSC). The lower the RSC, the more similar the horizontal transition rates are forced to be. Optimisations were terminated after a predefined compute time (Materials and Methods). Only the parameter sets for which f < 1 are shown, and their number for each RSC value is given underneath the corresponding set of points. (B) Input-output responses corresponding to the parameter sets obtained from optimisation with RSC = 0.005, for which f < 1 and increases with x. (C) Values of and in the parameter sets corresponding to the responses in B (RSC = 0.005). (D) Parameter values corresponding to the responses in B (RSC = 0.005). The green line represents the best parameter set (minimum f). Each parameter is plotted in units of , except for k_on, which is in units of . (E) Heatmap of the coupling score, f, with respect to and , with the other parameters set to the most optimal parameter set (green curve in D), along with select choices of and (crosses) whose corresponding input-output curves are shown in F. (F) Input-output curves corresponding to the parameter sets indicated in E. (G) Overlap between the concentration ranges over which the input-output curves in B change by 90%. Given two intervals [a,b] and [c,d], the overlap is computed as . The closer this value is to zero, the less overlap there is between the concentration ranges over which and exhibit the greatest change. (H) Values of and in the best parameter set for each choice of RSC.

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When examining the optimal parameter sets with a coupling score of f < 1 for RSC = 0.005, we found that and lie in the incoherent space, with and (Fig 4C). To confirm the relevance of the incoherent regulatory mode, we identified the best parameter set from this ensemble of optimization results (Fig 4D, green curve) and computed the coupling score while varying and within the range (Fig 4E shows a portion of this parameter space region; the green cross corresponds to the best parameter set). As expected, in this scenario, moving away from the identified minimum towards leads to a rapid increase in the coupling score, suggesting significant sensitivity of the coupling score to the value of . To observe this more directly, we computed the responses for the parameter points corresponding to the crosses in Fig 4E (Fig 4F). When we set (black cross), we observed that decoupling was significantly reduced, with ; and when we entered the coherent space by setting (pink cross), we found that exceeds (Fig 4F).

By analysing the curves in Fig 4B, we found that the concentration range over which the steady-state level changed the most was systematically different from that for the mFPT, which can be quantified by the overlap of the input ranges over which the curves change the most (Fig 4G, see also the green curves in Fig 4F). Examining the parameter sets, we noticed that the concentration, x_1/2, at which the steady-state level is half-maximal, i.e.,

is always close to , which is reminiscent of a Michaelis–Menten kinetic scheme (S3A Fig, left). We also observed that the concentration at which the normalised mFPT is minimised, which we denote by x_fast, is close to (S3A Fig, right). Together, this suggests that we can modulate the overlap to some extent by tuning k_off. Indeed, we found that increasing the value of k_off in the best parameter set in Fig 4D shifted the normalised steady-state curve rightward and increased x_1/2, while only minimally affecting x_fast (S3B Fig). However, we also found that increasing k_off beyond a certain critical value also increases (S3B-C Fig). This illustrates that, within an appropriate range of values of k_off, we not only achieve global decoupling in the sense that the variation in the steady-state is much larger than that of the activation time, but we can also achieve a concentration-dependent form of decoupling, in which and both vary with x, but over largely non-overlapping ranges.

Regarding the effect of the RSC value and thus the similarity of the rates of the various transitions, we found that, as we increased RSC to allow for rate scale separation (RSC ≥ 1), the optimal value of approached 1, whereas the optimal value of remained similarly large (Fig 4H). In other words, we observed a transition from an incoherent regime, in which , , and the transition rates are more tightly constrained, to a regime in which is unregulated (), , and the transition rates are separated (i.e., case 1.III in Table 1, Fig 3E).

We also noticed that the optimisations tended to yield values of r near the maximum possible value (r ≈ 10⁴, Fig 4D), although there was a spread of values. To ascertain whether a large value of r is necessary for decoupling in this context, we also ran optimisations with the additional constraints that and , so that the production transition proceeds on the same timescale as the preceding forward transitions, and (S4 Fig). As before, we observed the strongest decoupling when the ligand operates in an incoherent regime, with comparably low coupling scores as in the previous optimisation (S4 Fig). This suggests that decoupling due to incoherent regulation does not require a large value of r relative to the other transition rates.

Finally, we also considered regulatory regimes in which one or both of the backward transitions, and , are regulated. Specifically, we performed additional optimisations with the constraints and , but with the following choices of regulatory factors that may differ from 1: (I) and/or ; (II) and/or ; or (III) and/or (S6 Fig, S7 Fig, and S8 Fig, respectively; Appendix E in S1 Text). Here, we found instances of decoupling for cases II and III in the incoherent space ( and for case II, and for case III), in which the ligand-bound transitions were biased towards B₁ and B₃. This again demonstrates that incoherent regulation can give rise to decoupling. On the other hand, no instances of decoupling were found for case I, for which the reasons remain elusive (S6 Fig).

In summary, our findings demonstrate that, when the transition rates are similar to each other, incoherent regulation can facilitate decoupling between the steady-state level and activation time. In this regime, unlike decoupling due to rate scale separation, the activation time exhibits a more significant dependence on the input concentration, but this variation can occur over a concentration range that is largely non-overlapping with that over which the steady-state changes (Fig 4G), effectively leading to decoupling. We have also confirmed that these results extend to the larger graph, (S9D-E Fig).

Decoupling from an equilibrium of initial states

So far, we have defined the activation time in the ladder models, , as the mFPT from one initial state, U₁, to the terminal state, M, in which a copy of the readout M has been produced. This definition is reasonable in the setting of morphogen-mediated gene regulation in developmental systems such as the Drosophila blastoderm, a key model system that motivated our analyses in this paper [9,11,14,16]. In this system, the nuclei divide every few minutes, with each mitosis resulting in the repression of transcription and the condensation of chromatin into a broadly inaccessible state. Therefore, it is reasonable in this context to assume that, upon the initiation of a new nuclear cycle, the regulatory DNA element that binds the morphogen has been “reset” to exist in the state U₁.

However, in other contexts, such as those in which the cell is non-dividing or exhibits a long division time, it is plausible that the system exists in an equilibrium of initial states before the ligand is introduced. In this case, a more appropriate measure of the activation time would be the average mFPT to the terminal state over all possible initial states, each weighted by its steady-state probability (Fig 5A). In other words, we define,

(25)

where we have used the same notation as in Eq. 7, and is the steady-state probability of U_i in prior to introduction of ligand (x = 0). A mathematical justification of this definition is provided in Appendix F in S1 Text.

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Fig 5. Decoupling from an equilibrium of initial states.

(A) Schematic of the definition of activation time. This definition assumes that, before the ligand is introduced, the system has reached a steady state over the unbound states. That is, at the moment immediately prior to the introduction of ligand (t = 0 and x = 0), the system may occupy any of the unbound states (red vertices), each according to its steady-state probability. (B) Distributions of the coupling score, f, obtained from optimization with various rate scale constraints (RSC). Only the parameter sets for which f < 1 are shown, and their number for each RSC value is given underneath the corresponding set of points. (C) Input-output responses corresponding to the parameter sets obtained from optimization with RSC = 0.005, for which f < 1 and increases with x. (D) Values of and in the parameter sets corresponding to the responses in C (RSC = 0.005). (E) Parameter values corresponding to the responses in C (RSC = 0.005). The green line represents the best parameter set. Each parameter is plotted in units of , except for k_on, which is in units of . (F) Heatmap of the coupling score, f, with respect to and , with the other parameters set to the most optimal parameter set (green curve in E), along with select choices of and (crosses) whose corresponding input-output curves are shown in G. (G) Input-output curves corresponding to the parameter sets indicated in F. (H) Values of and in the best parameter set for each choice of RSC.

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To determine whether decoupling can be obtained with this alternative definition of activation time, we ran the constrained PSO with different choices of RSC (Eq. 24), again focusing on the case where and may be distinct from 1, and . As with the preceding analysis, we found that the strength of decoupling decreases as we decrease the RSC (corresponding to a stronger constraint on the transition rates), but we still observed significant decoupling at RSC = 0.005 (Fig 5B). As before, the increase in the coupling score with the RSC can be attributed to a smaller dynamic range in the steady-state level (Fig 5C, top). In this case, we found that the activation time shows little to no dependence on the ligand concentration, in contrast to our previous analysis (Fig 5C, bottom; compare to Fig 4B, bottom). We confirmed this observation with Gillespie simulations, which we performed to further validate the definition in Eq. 25 (S5 Fig).

Importantly, the optimized parameter sets were found to lie in the incoherent space for RSC = 0.005 (Fig 5D). Upon varying the regulatory factors, and , in the best parameter set (Fig 5E, green curve), we again found that the strongest decoupling is indeed obtained when and (Fig 5F and G). Moreover, increasing RSC results in optimal parameter sets that satisfy and (Fig 5H), again demonstrating that, as we allow for rate scale separation, we reach regulatory regimes in which only is regulated (case 1.III in Table 1). Finally, we confirmed that decoupling under incoherent regulation with respect to can also be observed in the model (S9F–G Fig).

Sensitivity of the proposed mechanisms to single-parameter variations

Our analyses demonstrate that decoupling can arise from two mechanisms: rate scale separation and incoherent regulation under rate scale constraint. We now examine how sensitive these mechanisms are to variations in individual parameters. For each optimal parameter set, , with steady-state response and activation time , we generated a perturbed parameter set, , in which the i-th entry was perturbed by setting its value to , where was randomly sampled from the uniform distribution on , and every other entry was held fixed. We then calculated the corresponding steady-state dynamic range, , and activation time dynamic range, , and compared these values with their unperturbed counterparts, as

(26)

We emphasize that, in this procedure, each parameter is perturbed one at a time; as such, this analysis does not reveal how the system may respond to simultaneous perturbations of multiple parameters. We expand on this point in the Discussion.

Fig 6A–6C show the results of this analysis for the rate scale separation mechanism. Here, was set to the best parameter set in Fig 3E (green line), which is an example of a case 1.III mechanism, in which the ligand regulates the transition via the regulatory factor (Fig 3A). This parameter set is shown in Fig 6A. For each parameter i, we sampled 1000 parameter sets and calculated the corresponding distributions of (Fig 6B) and (Fig 6C). As shown in Fig 6B, we found that is minimally affected by these perturbations. This suggests that, as long as the perturbation does not significantly decrease the rate scale separation, is minimally affected. As for , we found that the same parameter perturbations led to small changes in , with the largest changes arising from the second set of transitions (Fig 6C). In particular, we note that increasing (resp. decreasing) leads to an increase (resp. decrease) in . This was previously suggested by our theoretical analysis of the case 1.III mechanism (Table 1), although we note that this preceding analysis enforced the additional assumption that and .

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Fig 6. Sensitivity of decoupling via the rate scale separation (A–C) and incoherent regulation (D–I) mechanisms to single-parameter perturbations.

(A) The best parameter set, , obtained from the optimization in Fig 3A–E (see also Fig 3E, green line). (B–C) Distributions of (B) and (C) arising from perturbations in each parameter, starting from the choice of in A. The colormap represents the log-ratio of the perturbed parameter value, , with respect to the optimal parameter value, . (D) The best parameter set, , obtained from the optimization in Fig 4 (see also Fig 4D, green line). (E–F) Distributions of (E) and (F) arising from perturbations in each parameter, starting from the choice of in D. (G) A modified version of the parameter set in D, in which all horizontal rates have been set to the same value, , as described in the text. (H–I) Distributions of (H) and (I) arising from perturbations in each parameter, starting from the choice of in G. Here, the horizontal rates were perturbed as a single parameter, ℓ, as described in the text.

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In Fig 6D–F, we show the results of two similar analyses for the incoherent regulation mechanism. Here, was first set to the best parameter set in Fig 4D (green line), in which the horizontal transition rates were tightly constrained, with an RSC value of 0.005; this parameter set is shown in Fig 6D. Now, since introducing parameter perturbations can cause these constraints to be violated, and incoherent regulation is only useful for decoupling for tightly constrained rates (Fig 4H), we hypothesized that perturbing the horizontal transition rates in would lead to weaker decoupling. This is indeed what we observed: perturbing , , or in either direction resulted in significantly larger values of (Fig 6E). In contrast, we observed no such effect upon perturbing ; we attribute this to the significantly larger value of r in (Fig 6D), which is sufficiently large that a 10-fold increase in does not appreciably increase the probabilities of the backward transitions, and , relative to the probabilities of the terminal transitions, and . Moreover, we found that perturbing the regulatory parameter, , also gives rise to increased , but perturbing does not; this is consistent with what we observed in Fig 4E, where a thin sliver of values in in the incoherent space affords the strongest decoupling under strong rate scale constraint. As for , we found that its sensitivity to parameter perturbations is comparable to that for , with the largest changes arising from perturbations to the horizontal transition rates (Fig 6F).

We then sought to understand whether the incoherent regulation mechanism is sensitive to a different type of perturbation, in which we force all horizontal rates to be equal (Fig 6G–I). Here, we set to the same optimal parameter set as in Fig 4D, but with , , and set to the value. This modified parameter set is shown in Fig 6G. We then treated all four of these parameters as one common parameter, denoted ℓ, which we perturbed in the same way as before. Here, since perturbing ℓ does not change the ratios between the horizontal rates, which are fixed at 1, we hypothesized that the decoupling mechanism corresponding to is more robust to such perturbations. Again, this is indeed what we observed (Fig 6H). We also observed a similarly high sensitivity to perturbations in and low sensitivity to perturbations in as in the preceding analysis. Finally, we found that perturbing ℓ also leads to negligible changes in (Fig 6I), in contrast to the sensitivity of to perturbations in the horizontal transition rates in Fig 6F. This is consistent with our observations in Appendix C in S1 Text, on the effects of variable horizontal rates on . In all, these results demonstrate that rate scale separation is less sensitive to single-parameter variations than incoherent regulation with a rate scale constraint, due to the intrinsic requirement that the horizontal rates be similar in the latter mechanism.

In S10 Fig, we show the results of similar analyses for two parameter sets identified from the optimizations in Fig 5, which were performed with respect to the definition of activation time (Eq. 25). As described next, these analyses yielded very similar results to those shown in Fig 6, corresponding to the definition.

First, we set to the best parameter set obtained for the optimization with RSC = 3 (Fig 5B), in which the rates are not tightly constrained, and the two forward regulatory factors, and , may be distinct from 1 (S10A Fig). We observed significant rate scale separation in this parameter set and a value of (Fig. 5H). To make this analysis comparable to that in Fig. 6A–C, we fixed and did not subject it to perturbations. We found that perturbing the other parameters as before yielded very similar distributions to the analysis in Fig 6C (S10C Fig). As for , we observed qualitatively similar results to the analysis in Fig 6B, but with slightly greater sensitivity to perturbations in and (S10B Fig). We then set to the best parameter set obtained for the optimization with RSC = 0.005 (Fig 5E, green line, and S10D Fig), in which the rates are tightly constrained, similar to the best parameter set in Fig 4D analysed in Fig. 6D–F. Again, we found that parameter perturbations yield a very similar pattern of distributions to the corresponding analysis in Fig 6F (S10F Fig), and a qualitatively similar pattern of distributions to the analysis in Fig 6E (S10E Fig). Finally, we set to the same parameter set but with , , and set to the value, and treated all four parameters as one common parameter, ℓ (S10G Fig), akin to our previous analysis in Fig 6G–I. Again, the resulting and distributions (S10I Fig and S10H Fig, respectively) were similar to the corresponding distributions in Fig 6I and H.

Finally, we asked how this picture changes when we constrain the parameters to better reflect plausible scenarios in biological input-output systems. Specifically, we used our optimization procedure to identify parameter sets that achieve a prescribed minimum value for over a reduced input concentration range, as well as a target value for along that range, while also achieving strong decoupling globally.

We chose a reduced input concentration range spanning three orders of magnitude,

(27)

which is a plausible estimate for spatial morphogen gradients in developing embryos [42,43]; and sought to achieve a steady-state dynamic range that exceeds half its theoretical maximum, i.e.,

(28)

We also chose a target value of , so as to attain an activation time that roughly scales with the half-life of the readout, M; this is again broadly consistent with measurements in gene regulation systems, in which both the activation time and the half-life of the produced mRNA are often on the order of minutes [44,45]. Since can change with x, we sought to enforce this target value by implementing the constraint,

(29)

That is, we accepted values within 10% of the target value. We first ran the optimization procedure with these three new constraints, to obtain a parameter set that achieves strong decoupling through a rate scale separation mechanism (S11A Fig). We then ran the optimization procedure with these three constraints, plus the additional constraint that

to attain a parameter set that achieves strong decoupling with significant rate scale constraint (S11B Fig).

We then performed the same sensitivity analysis as described above on these new parameter sets. Here, we computed and over the reduced concentration range, and we also computed the change in at x = 10⁻³ c.u. due to each parameter perturbation, which we defined as

(30)

For the rate scale separation mechanism, we found that only perturbing the slow forward rate, , incurs large changes in (S11C Fig, left), which is consistent with the intuition that the corresponding transitions, and , are rate-limiting. When the horizontal rates are constrained to be equal, we found that perturbing the one common rate, ℓ, incurs similarly large changes in (S11D Fig, left).

Meanwhile, we observed largely similar patterns of changes in to the corresponding parameter sets in Fig 6A–C and 6G–I, except for a few parameters. First, we found that, for the rate scale separation mechanism, is more sensitive to perturbations in and than in the previous analysis (compare Fig 6B with S11C Fig, center). This may be due to the enforcement of a target value for (Eq. 29), which yielded an value of for the parameter set in S11A Fig, as opposed to a value of in the corresponding parameter set obtained without this constraint, in Fig 6A. Second, we found that, for the incoherent mechanism, is much less sensitive to perturbations in the regulatory factor, , than in the previous analysis (compare Fig 6H with S11D Fig, center). In S11E Fig, we show that perturbing actually gives rise to variation in outside the concentration range defined in Eq. 27 (compare with the unperturbed curve in S11B Fig). We also note that, in S11B Fig, the range of x over which changes does not overlap with the corresponding range for , akin to the optimal parameter sets in Fig 4B (see Fig 4G). These results suggest that this more local form of decoupling is less sensitive to perturbations in . Finally, we observed mostly similar patterns of changes in as in the previous analysis (compare Fig 6C and I with S11C Fig, right, and S11D Fig, right, respectively).

In all, these findings suggest that imposing the additional constraints we have introduced in Eqs. 27–29 can alter the sensitivity patterns of decoupling to single-parameter variations, but in a manner such that decoupling is only sensitive to perturbations outside the concentration range of interest. We also find that the strength of decoupling over the reduced concentration range is still less sensitive than other features, such as the activation time itself over the same range.

Calculating the steady-state level, activation time, and their dynamic ranges

As described in the section “Modeling approach and mathematical setup,” we used both numerical and analytical approaches for calculating the steady-state level, activation time, and their dynamic ranges, which we defined using the graph-theoretic linear framework [17–20] (Appendix A in S1 Text). For the analytical calculations, we implemented a symbolic version of the Chebotarev–Agaev recurrence (Eq. 14), using the Python package SymPy [72]. Barring additional symbolic simplifications, this approach allows us to obtain exact algebraic expressions for the steady-state level and activation time for any graph, G, in O(n⁴) arithmetic operations, where n is the number of vertices in G. This is much more efficient than any method based on direct enumeration of the spanning trees and/or forests that feature in Eqs. 9 and 10, which can scale exponentially with n [19].

For the numerical calculations, we calculated the steady-state level by obtaining the SVD of , identifying the right singular vector corresponding to the zero singular value, and normalizing appropriately. This right singular vector is unique whenever G is strongly connected. The normalized vector, which is the steady-state vector, p^*, arising from the master equation (Eq. 1), was then used to evaluate , as per Eqs. 4 and 15. On the other hand, we calculated the activation time by obtaining the QR decomposition of the left-hand matrix in Eq. 8, and using the corresponding solution vector to Eq. 8 to evaluate , as per Eq. 15. Briefly, a QR decomposition of an invertible matrix is a decomposition, , such that Q is orthogonal, , and R is upper-triangular; such decompositions are useful for solving matrix equations of the form . While the exact runtime complexities of these decompositions can differ between implementations, we expect that they will be asymptotically faster or comparable to the O(n⁴) runtime of the Chebotarev–Agaev recurrence, and—more importantly—faster in practice due to the availability of highly optimized implementations.

All calculations were implemented in C++, using multiple-precision floating-point numbers from the Boost.Multiprecision library [73] with a precision of 100 digits, and implementations of SVD and QR, from the Eigen library [74]. Python bindings were implemented to call these C++ functions from Python, so that they could interface with the PyMoo optimization suite (see below).

We checked that the mFPTs obtained from Eq. 13 agree well with estimates obtained from Gillespie simulations [32,75] on a simple four-vertex graph (S1A Fig). We sampled 100 combinations of values for the rates, each from a log-uniform distribution on the range , and ran increasing numbers of Gillespie simulations starting from vertex 1 to estimate the mFPT to vertex 4. As expected, we observed that the agreement between these estimates and their corresponding exact values, given by Eq. 13, increases with the number of trajectories (S1B Fig).

To numerically compute the dynamic range for a given parameter set, we calculated and over a logarithmic concentration range of , with a logarithmic stepsize of . We chose this wide range to ensure that these calculations capture the complete dynamic range of these outputs. For each output, we then identified the maximum and minimum value over this concentration range, and computed the dynamic ranges according to Eq. 16.

Minimization protocols for the decoupling score f

In preliminary investigations for this study, we compared various metaheuristic optimisation algorithms to find parameter sets that exhibit decoupling. We found the Particle Swarm Optimization (PSO) implementation provided by PyMoo [38], a multi-objective optimization suite, to be the most efficient option, and therefore used it for all the numerical analyses discussed in this work.

The PSO algorithm was first introduced by Kennedy and Eberhart in 1995 [37]. Conceptually, the procedure is based on a swarm of particles, each with an associated position and velocity vector, which are iteratively updated to minimise an objective function, f. We denote the position and velocity of the i-th particle along the d-th dimension at time t as and , respectively. (Here, t is treated as an integer variable.) In our case, each particle corresponds to a parameter set, with the particle’s position given by the parameter values. Meanwhile, the velocity of the i-th particle along the d-th dimension, , is determined by:

1. that particle’s position at which f attained its lowest value throughout the particle’s trajectory up to time t, which we denote by ; and
2. the position at which f attained its lowest value throughout the entire swarm’s history up to time t, which we denote by G_d(t).

More specifically, is given by

where is an inertia factor, are noise coefficients representing the level of “craziness” in the optimisation, and c₁ and c₂ balance the contributions from the i-th particle’s “personal” behaviour () and the swarm’s global behaviour (G_d) [37]. PyMoo dynamically adjusts , c₁, and c₂ throughout the optimisation, with initial values of and , following the prescription outlined in [76]. Finally, the position of each particle is updated as

We ran PSO for each of the problems defined in the paper, passing a total of 100 different random seeds to obtain a population of optimal solutions. To sample the initial set of particles, we used Latin hypercube sampling, which is the default choice in PyMoo. To enforce parametric bounds, the parameters that fall outside the defined bounded range are set to the closest bound value during the optimisation.

As the termination criterion for each PSO run, we used either convergence to a score of f < 0.1 for more than five consecutive generations, or a computation time exceeding 23 hours. The termination criterion that was used for each PSO run is specified in the corresponding figure caption.

Optimisations were performed on the O2 High Performance Compute Cluster at Harvard Medical School and the Scientific Computing Core Facility of the Department of Medicine and Life Sciences at Pompeu Fabra University.