An elementary flux mode-based approach for constructing weakly reversible and deficiency zero network translation

One of the improvements made to resolve a key issue of COMPILES is the replacement of its network translation scheme with a more efficient procedure. In [5], a computational method was developed to construct a WR and DZ translation via an elementary flux mode-based approach. Elementary flux modes (EFMs) are flux-balanced pathways in the network that cannot be simplified in the sense that it is not possible to remove a subset of active reactions from an EFM and still be able to build a flux-balanced path using only the remaining active reactions [5,6] (see the Methods section for more details).

The approach proposed in [5] consists of three main steps:

1. From the given CRN, compute the set of EFMs .
2. Construct a reaction-to-reaction graph which is common source compatible (CS-compatible) and elementary flux mode compatible (EM-compatible) with the given CRN.
3. Determine the translation complexes to produce the WR and DZ network translation.

To illustrate the process, we consider the histidine kinase system below [21]. Here, the histidine kinase (X) can autophosphorylate () and can transfer the phosphate group to a response regulator (Y) yielding (), which undergoes autodephosphorylation.

Step T1. Since EFMs represent flux-balanced pathways in the network, it is necessary to track the net production and consumption of species in the CRN. This information is codified in the stoichiometric matrix N (see the Methods section for more details on the basics of chemical reaction networks). Given this, an EFM is a flux vector that satisfies Nv = 0 together with the nonnegative flux constraint . The central mathematical constructs that describe the space of all flux vectors that satisfy Nv = 0 where (i.e., admissible flux vectors) are polyhedral cones [6].

By defining , we can observe that the set

is a polyhedral cone that contains exactly the admissible flux vectors [6]. Moreover, it has been established in [6] that the extreme rays of this polyhedral cone correspond to the EFMs of the network. Alternatively, the set of EFMs is the minimal generating set of P.

The general method that computes this minimal generating set or the extreme rays of P is called the Double Description Algorithm. Several variations of the algorithm exist. In this paper, we use the Canonical Basis Method, which exploits the structure of the matrix A as defined above [6]. The outline of the method is presented in Algorithm 1.

Algorithm 1 Computing EFMs

Input: Stoichiometric matrix N

Output: A matrix R whose columns are the EFMs

1. Initialize: Set k = r + 2, , and .

2. Prepare rays in R_old for partitioning: While k < r + 2m + 2, do:

  a. Select a row v of N not in A_old.

  b. Update .

  c. Let , and where q is the number of columns of R_old.

3. Partition rays in R_old: Let , , and .

4. Generate extreme rays: For each , do:

  a. Compute .

  b. Test for extremeness. If is an extreme ray, append to R_new.

5. Update R_old and R_new:

  a. Remove columns of R_new indexed by .

  b. Update: , , and k=k+2.

6. Repeat: Go back to Step 2. Stop when no row of N remains unchosen.

Going back to the histidine kinase system consisting of four reactions, we can define its stoichiometric matrix N as follows

Applying Algorithm 1, the output R is given by

which corresponds to the set of EFMs . See the Supplementary Information for the step-by-step procedure in the computation.

Step T2. Once the EFMs are computed, a reaction-to-reaction graph is then constructed that must be both CS-compatible and EM-compatible with the given CRN (see Theorem 3 in the Methods section).

A reaction-to-reaction graph is a directed graph in which the reactions of the CRN are treated as the vertices of the graph [5]. The CRN is said to be CS-compatible whenever reactions of the following form in the CRN

forces if and only if , where is the edge set of the reaction-to-reaction graph [5]. Moreover, both the CRN and the reaction-to-reaction graph are said to be EM-compatible if every EFM of the CRN corresponds to the vertices of a minimal directed cycle in the reaction-to-reaction graph, and vice versa [5].

In [5], a binary linear programming (BLP) problem was formulated to construct the desired reaction-to-reaction graph while enforcing CS compatibility and EM compatibility with the given CRN. In our histidine kinase system, the BLP model results in the reaction-to-reaction graph shown below. See the Supplementary Information for the detailed discussion.

Step T3. The final step of the translation procedure is to determine the translation complexes to obtain a WR and DZ network translation. Intuitively, translation complexes are complexes added to individual reactions in the translation process. This modifies the graphical structure of the CRN without changing its stoichiometric matrix [18]. This is crucial to preserve the dynamics of the system.

Theorem 3 provides the method for determining the translation complexes, which involves solving the linear system in Eq (9). A more efficient procedure, however, is proposed in [5] to determine the translation complexes instead of solving the linear system in Eq (9) directly.

The approach is similar to a graph traversal: once a value is fixed at a single node, the remaining values are computed incrementally using Eq (9) by traversing the directed edges of the reaction-to-reaction graph. The outline of the procedure is shown in Algorithm 2, which is based in [5]. This strategy exploits the structure of the reaction-to-reaction graph to propagate the values of the translation complexes step-by-step. This is more efficient than solving the full matrix system which may have redundancies that can lead to unnecessary computational overhead.

Algorithm 2 Determining the translation complexes

Input: Edges of the reaction-to-reaction graph, matrix of source complexes and product complexes , linkage classes

Output: Translation complexes

1. Check consistency of the linear system. Determine whether rank(A) = rank(B), where B is the augmented matrix , and A and b come from Eq (9). Exit algorithm if .

2. Seed the process. Set . Select an arbitrary and set . Move to the “active” list .

3. Expand. For every in the “active” list, choose a neighbor , and then calculate based on Eq (9). Move to the “processed” list .

4. Propagate. Once we have checked all neighbors of the current “active” list , set . Set and repeat Step 3.

5. Check for completion. If the “active” list is already empty but , go back to step 2. If , exit the algorithm.

6. Update complexes. Add the computed translation complexes to reaction i.

7. Ensure nonnegative complexes. For each linkage class, shift complexes with negative entries to ensure that the stoichiometric coefficients of species per complex is nonnegative.

In our running example, the histidine kinase network consists of four reactions, and so four translation complexes must be determined. Applying Algorithm 2, the translation complex for each reaction is given by

which results in the following translated network which is guaranteed to be WR and DZ by Theorem 3. See the Supplementary Information for the step-by-step procedure in the computation.

It should be noted that a WR and DZ translation from the method of Theorem 3 is not unique. For instance, translating using the translation complexes , , and yields a different network but is still WR and DZ. This arises from Steps 2 and 3 of Algorithm 2, where different choices of from may lead to different outcomes.

In the next subsection, we demonstrate the enhanced computational framework for deriving analytic steady states using a two-protein gene transcription model. We use a different example to better illustrate the framework. The process involves decomposing the network into its independent subnetworks and subsequently merging translated versions of the subnetworks. The histidine kinase example does not admit a nontrivial independent decomposition, so it does not demonstrate these decomposition and merging steps.

Demonstration: An enhanced framework for deriving the analytic steady state of a two-protein gene transcription model

We now demonstrate the enhanced framework for deriving analytic equilibria by considering a two-protein transcription model, which is a mass-action system consisting of ten reactions that describe a gene transcription motif involving two proteins P₁ and P₂, each produced by their respective genes G₁ and G₂ [7,21]. Table 1 shows the reactions of the network grouped according to their functional roles.

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Table 1. The reactions of the two-protein gene transcription model and their descriptions.

https://doi.org/10.1371/journal.pcbi.1014633.t001

The first step of the procedure is to break down the network into its independent subnetworks as shown in Fig 1a. The rank of the entire network is given by

(1)

while the ranks of the subnetworks are as follows

Since , the decomposition in Fig 1a is an independent network decomposition. Among the five subnetworks, three are already WR and DZ, whereas the first two require network translation to satisfy these conditions. Following the three-step translation scheme outlined in the previous subsection, we use G₁ as the translation complex for the third reaction, while leaving the first reaction unchanged. Refer to Fig 1b. As a result, the third reaction becomes , which is the reverse of the first reaction. The translated subnetwork with reversible reactions defines the stoichiometric CRN, denoted by , associated with . Clearly, it is both WR and DZ .

Next, we construct the kinetic-order CRN, which is denoted by , associated with . To do this, we preserve the reaction rates of the original network so that the translated network remains dynamically equivalent to the original system (see Methods section under Generalized chemical reaction networks for a more detailed discussion). In particular, we assign the reaction rate to the translated reaction because P₁ is the source complex of the corresponding reaction in the original network. In this manner, we preserve the reaction kinetics of the original reaction. Moreover, the reaction vector of the translated reaction is given by

which is precisely the third column in the stoichiometric matrix of the original network (see Eq (1)). These guarantee that the translated network generates the same dynamical system as the original network. Otherwise, the translated network would no longer be dynamically equivalent to the original network, and the subsequent steady state analysis would not correspond to the original biochemical system.

We then take all the edges of while assigning to each of the source nodes of the edges the corresponding source complexes from the original network. Since P₁ is the original source complex of the third reaction, two complexes become associated with a single node in . We separate them using a phantom edge, defined as an edge equipped with a free parameter () that serves as its “rate constant”, in such a way that the resulting network is WR. For a phantom edge, we always get a zero stoichiometric vector, since both the head and tail of the edge correspond to the same complex in the stoichiometric CRN. This construction yields the kinetic-order CRN , which is both WR and DZ . The stoichiometric CRN and kinetic-order CRN constitute the translated network for . Since both and are WR and DZ, the translated network is WR and DZ as well.

Similarly, we apply the procedure to using G₂ as the translation complex for the fourth reaction while keeping the second reaction unchanged. This yields a translated network that is both WR and DZ as shown in Fig 1b.

The next step is to merge the subnetworks into a single unified network instead of treating each subnetwork as an individual piece. This step is supported by Theorem S1 in the Supporting Information, stating that the preservation of weak reversibility and zero deficiency under merging of independent translated subnetworks possessing these properties is guaranteed. In the two-protein gene transcription example, Fig 1c illustrates how the subnetworks are merged. This is simply done by taking the union of the reaction sets of the translated subnetworks. Since the subnetworks do not share any common complexes, the merging process preserves the interconnections within each subnetwork, effectively treating them as components of a single larger network. It can be easily verified that the resulting merged network is both WR and DZ. In particular, both the stoichiometric CRN and kinetic-order CRN of the merged network are WR and DZ. This is also guaranteed by Theorem S1 in the Supplementary Information.

Finally, we compute the equilibrium parametrization using the full network rather than treating each subnetwork individually. We refer the readers to [2,22] for a detailed discussion on equilibrium parametrization. Fig 1d shows the final equilibrium parametrization of the two-protein gene transcription model. It can be observed that P₁ and P₂ are free parameters in the parametrization. If we set particular values for the rate constants and free parameters P₁ and P₂, we get the steady state concentration values of the species G₁, G₂, D, C₁, and C₂.

Improvements in CRITERIA over COMPILES

In this subsection, we summarize the improvements introduced in CRITERIA over COMPILES. Specifically, we incorporate three key modifications to the COMPILES framework to address its issues and limitations. First, we adopt the method developed in [5] to translate networks into becoming WR and DZ using EFMs. This was discussed in detail in a previous subsection using a simple histidine kinase example. By replacing the original translation procedure in COMPILES, we resolve a major computational bottleneck making CRITERIA more practical for analyzing larger reaction networks.

Second, we reorder the steps of COMPILES. Instead of computing the equilibrium parametrization of each subnetwork individually, we first merge all subnetworks into a single network by taking the union of their reactions, and then compute the parametrization for the entire network. This modification is crucial for avoiding dependencies in the equilibrium expressions, as described in the Introduction section. To illustrate this issue more clearly, we consider again the MAPK example in the Introduction. The parametrization obtained by COMPILES for the species and is given by

There is an interdependency issue in this example because the two parametrization equations depend on each other in a circular way, so that none of the involved species can be expressed solely in terms of the free parameters and rate constants without further algebraic manipulation. If we substitute the equation for to that of , we arrive at the equation

which is a hidden algebraic condition on the parameters. However, after observing this relation, substituting the parametrization into the ODE system does not make all time derivatives vanish indicating that the output COMPILES obtained is incorrect. This is what we aim to address as far as the interdependency issue is concerned.

Finally, we extend the applicability of the parametrization procedure to handle cases in which the deficiency of the kinetic-order CRN is positive based on the theory established in [22]. In particular, the core algorithm in COMPILES does not consider the theoretical foundations for models with positive kinetic deficiency according to Case 2 of Theorem 2 (see the Methods section). This means that COMPILES neither processes nor outputs the additional algebraic conditions necessary for the final parametrization of networks with positive kinetic deficiency. Take, for example, the MAPK model considered above. Its kinetic deficiency is equal to one. By Case 2 of Theorem 2, one additional condition needs to be imposed in the final parametrization. Applying CRITERIA, we get

as the additional condition which is not captured by COMPILES. By considering Case 2 of Theorem 2, we get a valid final parametrization for models whose kinetic deficiency is positive. In the next subsection, we compare the runtime performance of COMPILES and CRITERIA by applying them to a variety of models.

Runtime performance comparisons

We apply CRITERIA to 26 models and evaluate its performance by comparing its runtime with that of COMPILES. We also assess the performance gains of the new network translation scheme by comparing the EFM-based translation with the translation method used in COMPILES. Computational experiments were conducted on a MacBook Air equipped with an Apple M2 processor and 8 GB of RAM. All runtime measurements were obtained under identical hardware and software conditions.

To contextualize performance results, Table 2 summarizes the network properties of the benchmark models used for the comparisons. The models are listed in order of increasing number of reactions. It is worth noting here that all benchmark models are neither WR nor DZ.

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Table 2. Network properties of benchmark models with r = number of reactions, m = number of species, n = number of complexes, s = rank of the network, l = number of linkage classes, effective deficiency, and = kinetic deficiency.

https://doi.org/10.1371/journal.pcbi.1014633.t002

We impose a maximum runtime of 3600 seconds; any execution that does not complete within this time limit is terminated. For our computational experiments, we perform five independent runs, and report the mean and standard deviation of the resulting runtimes. In the first experiment, we compare the runtime performance of the EFM-based network translation, which we call REWARDZ (tRanslation via Elementary flux modes towards WeAkly Reversible Deficiency Zero networks), with TOWARDZ (TranslatiOn toward WeAkly Reversible and Deficiency Zero networks), the translation method used in COMPILES [23]. Table 3 presents this comparison. It should be noted that network decomposition has not yet been integrated in both computational frameworks. We make the following observations:

• REWARDZ successfully produced a valid translation for nearly all models, with the exception of Hernandez’s Influenza Virus Model and CRISPRi Model 3, whereas TOWARDZ succeeded in only five models. In particular, TOWARDZ failed to translate any model containing more than six reactions. It is important to note that the inability of TOWARDZ to generate the desired output within the prescribed maximum runtime is not caused by insufficient RAM. We monitored the memory usage of MATLAB throughout the computation for Model Z and observed that the peak real memory consumption is 209 MB which is relatively small compared with the 8 GB of available system memory of the machine. This suggests that its failure to obtain a valid translation is more likely attributable to the computational complexity of the translation search process.
• Among the models for which REWARDZ successfully constructed a WR and DZ translation, the runtime was under 3 seconds in all cases except for the full CRISPRi model (CRISPRi Model 4). This model, which is the largest considered in the paper, contains 42 reactions and has deficiency equal to 10.
• There is no significant difference in runtime among the models for which both methods successfully produced a valid translation, with the exception of the Hybrid Histidine–Kinase Model. This model, which consists of six reactions and ten species and has a deficiency of two, exhibited a markedly better performance under REWARDZ.

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Table 3. Comparison of running times (in seconds) between TOWARDZ and REWARDZ, and between COMPILES and CRITERIA. A maximum runtime of 3600 seconds is imposed for the computational experiments. Each entry represent the mean runtime standard deviation over five independent runs. Entries marked with “ 3600” indicate that the method was unable to produce the desired output within the time limit. Entries labeled “No valid translation” signify that the method completed within the prescribed maximum runtime but did not identify a WR and DZ translation. Lastly, entries labeled “Error” encountered an error in the code.

https://doi.org/10.1371/journal.pcbi.1014633.t003

The result in Table 3 demonstrates that REWARDZ offers a substantial improvement in terms of running time in generating a WR and DZ network translation, especially in larger models. It underscores the effectiveness and scalability of the enhanced computational framework in handling more complex biochemical systems.

We now proceed to compare the runtime performance of COMPILES and CRITERIA. Similar to the comparison above, we impose a maximum runtime of 3600 seconds. If steady states are not parametrized within the time limit, we exit the program. Moreover, we repeat the experiment five times, and the results are reported as the mean standard deviation across these runs. As shown in Table 3, COMPILES was unable to derive the analytic equilibria of several benchmark models. Of the 26 models, it failed to parametrize the equilibria of 11, whereas CRITERIA successfully obtained equilibrium parametrizations for all models. Notably, CRITERIA was able to solve the analytic equilibria of all models in less than 7 seconds except for the full CRISPRi model which required approximately 12 seconds. In addition, COMPILES encountered code errors in 2 models, namely the 3-site PD Network and CRISPRi Model 2. These results show the performance gains of CRITERIA over COMPILES.

It can be observed that COMPILES performed slightly better than CRITERIA in six models. This is because, in these models, network decomposition has broken down the network into WR and DZ subnetworks, eliminating the need for translation. To elucidate this, we consider the 2-Protein Gene Transcription Model, CRISPRi Model 1, and the Insulin Signaling Pathway. Fig 1 shows that the 2-Protein Gene Transcription Model consists of five independent subnetworks, three of which are already WR and DZ. The remaining two subnetworks that require translation each contain only two reactions. For the Insulin Signaling Pathway, five of the ten independent subnetworks are already WR and DZ. The remaining five subnetworks that require translation only have three reactions, four reactions, and two reactions in three subnetworks, respectively. Lastly, the CRISPRi Model 1 (smallest CRISPRi model considered in this paper) consists of eight independent subnetworks, none of which are WR or DZ, but each contains only two reactions.

Based on these results, COMPILES appears to handle smaller networks slightly more efficiently than CRITERIA. This is likely due to the fact that CRITERIA requires the solving of EFMs and a BLP, which can be computationally more demanding. For more details, see the Discussion section.

We verify the correctness of the output produced by CRITERIA by substituting the derived equilibrium expressions into the model’s ODE system using MATLAB. To get the mass-action ODE system of the models, we use CHECKMATE (https://github.com/pvnlubenia/CHECKMATE), which is a MATLAB code used to generate the ODE of a CRN with mass-action kinetics. We then plug in the output produced by CRITERIA into the ODE system. Indeed, the time derivatives evaluate to zero, confirming that the solutions are indeed valid equilibria.

In the next subsection, we apply CRITERIA to investigate key dynamical properties of selected biochemical systems such as multistationarity and ACR.

Application: Using equilibrium parametrization to examine ACR and multistationarity

We now apply our method to the EnvZ-OmpR system, a two-component signaling pathway consisting of the sensor kinase EnvZ, denoted by X, and the response regulator OmpR, denoted by Y [19]. In the network below that corresponds to Model L in Table 2, both X and Y exist in phosphorylated forms, denoted by and , respectively. The sensor X undergoes autophosphorylation through ATP (species T) binding and hydrolysis. The phosphorylated sensor catalyzes the transfer of the phosphoryl group to Y. Furthermore, X dephosphorylates using either ATP or ADP (species D) as a cofactor [19]. In the model, the effects of both ATP and ADP are simultaneously considered. Experimental observations indicate that the system exhibits concentration robustness, which has been analytically verified by demonstrating that the equilibrium concentration of depends solely on the rate constants. [14,19,22].

The reaction network of the system consisting of 14 reactions is shown below.

This can be decomposed into two independent subnetworks. The first subnetwork is given by

which is already WR and DZ. The second subnetwork, however, requires translation. It is shown below together with the translation complexes to achieve WR and DZ.

By applying CRITERIA, we obtain the following equilibrium parametrization

with free parameters and Y, and set of conservation laws

Observe that the equilibrium concentration of depends only on the rate constants confirming the result in [22]. Now, integrating the conservation equations with respect to t, we get

(2)

(3)

where T₁ and T₂ are nonnegative constants dependent on the initial conditions of the species involved in their respective equation. It is important to note that the conservation laws reported by CRITERIA are obtained by computing a basis for the left null space of the stoichiometric matrix of the reaction network [24,29,30]. Consequently, the reported conservation laws are not guaranteed to have positive coefficients as can be observed by the second conservation law of the EnvZ–OmpR model. This, however, does not affect the final equilibrium parametrization since conservation laws are not used in its derivation. In other words, the derivation of the equilibrium parametrization and the computation of conservation laws are independent procedures.

Therefore, the choice of a basis for the left null space of the stoichiometric matrix has no effect on the equilibrium parametrization or its positivity properties. In particular, replacing a reported conservation law with another valid conservation law with negative coefficients would not alter any of the steady state expressions generated by CRITERIA. Such valid conservation laws can always be obtained as linear combinations of the reported basis vectors. A detailed discussion on the computation of conservation laws is provided in the Supplementary Information.

To facilitate our computation, we express the rate constants in the equilibrium parametrization in terms of alphas so that , , , , , , and , where for i = 1,2,3,4,5,6,7. Plugging these expressions into Eq (2) and Eq (3), we get

(4)

(5)

From Eq (4), we have , where . Similarly, we can write Eq (5) as

where so that

(6)

(7)

The left hand side of the equation is a cubic polynomial in with two sign changes indicating that the equation can either have two or zero positive real solutions by Descartes’ Rule of Signs. Refer to the Supplementary Information for the proof. This means that can admit either two positive equilibria or none at all, implying that the system has the capacity for multistationarity. In other words, the system can have two positive equilibria for a particular choice of rate constants and initial conditions.

Alternatively, observe that the cubic expression in Eq (7) has a common factor . We can then write the cubic polynomial as , with g being quadratic. Setting , we have , where

We discard the case when since we are interested in looking at the positive equilibrium. When the discriminant of g is positive, an explicit parameter condition for the existence of two distinct equilibria is obtained. Suppose . From the Supplementary Information, it has been shown that A < 0, B > 0, and C < 0. We can infer from the Vieta’s formulas that the product of the roots of g is so that either both roots are positive or both of them are negative. By the quadratic formula, we have

Note that and . So, we have x₂ > 0 implying that x₁ > 0. Under the assumption, these two positive roots correspond to the two distinct and positive equilibria of the system. This behavior is supported by Fig 2 which shows representative cases for specific choices of rate constants and initial conditions. In particular, the first function exhibits two positive roots, whereas the second admits none.

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Fig 2. Graphs of two functions of the form with positive constants .

The x-axis represents the value of at steady state. The first function, shown in red, has two positive zeroes while the second function, shown in blue, has no positive zero.

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In the context of the EnvZ-OmpR system, the possibility for the existence of two positive equilibria or none at all provides a mechanistic explanation for how the EnvZ–OmpR pathway can implement threshold-based osmotic sensing. One equilibrium may represent a basal regulation, while the second may correspond to a fully activated stress-response state. This provides a rigorous mathematical foundation for its role as a master regulator of osmotic stress response. This result demonstrates the predictive power of CRITERIA over numerical simulations. While the latter can show specific cases of multistationarity, the analytical result provides a guarantee that no more than two positive steady states can exist regardless of the chosen rate constants. CRITERIA reveals how the network’s structure, rather than just specific parameter values, dictates these functions.

For our next application, we consider the full CRISPRi toggle switch model which consists of 17 species and 42 reactions [25]. This is Model Z of Table 2. In this system, a catalytically inactive protein (dCas9) forms complexes with two single-guide RNAs (sg1 and sg2). These complexes subsequently bind both to their target genes (G₁ and G₂, respectively) and also to the opposing genes [2,25]. Binding to the intended target genes is referred to as specific binding, whereas binding to the opposing genes is called unspecific binding. The analysis in [25] shows that the interplay between specific and unspecific binding of dCas9/sgRNA complexes to the genes explains bistability in the CRISPRi toggle switch model.

Here, we show that, in addition to the system’s capacity for multistationarity, the model also exhibits ACR in the dCas9 species. To establish this, we parametrize the equilibria of the system by applying CRITERIA. Fig 3 illustrates the decomposition, translation, and subsequent merging of the reaction network prior to deriving the equilibrium expressions. The species of interest in this analysis is X13, which represents the dCas9 protein. As shown in Fig 3b, the network has five independent subnetworks, two of which are neither WR nor DZ. Using the EFM-based network translation outlined above, we apply the translation complexes indicated on the right in each of the reactions in Fig 3b producing a generalized reaction network that is both WR and DZ.

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Fig 3. Network decomposition, translation, and merging of the full CRISPRi model for equilibrium parametrization.

a The reaction network consists of 17 species and 42 reactions. The species of interest in our analysis is X13, which represents the catalytically inactive dCas9 protein. The full correspondence between model species and network variables is provided in the Supplementary Information. b The network has five independent subnetworks, two of which are neither WR nor DZ. The other three subnetworks consist only of two reactions which are reversible. The complexes to the right of each reaction are the translation complexes after applying the EFM-based translation method described above. c Adding the translation complexes to each reaction yields the following stoichiometric CRN and kinetic-order CRN, which together constitute the generalized reaction network for the full CRISPRi model. The stoichiometric CRN is the resulting network after adding the translation complexes to each of the reactions in the original model. On the other hand, the kinetic-order CRN, which determines the kinetics of the system, consists of the original source complex of the reaction network before translation, thereby ensuring that the dynamics is preserved. The reactions shown in red in the kinetic-order CRN are referred to as phantom edges. They connect identical complexes in the stoichiometric CRN. In the example, both the stoichiometric and kinetic-order CRNs are WR and DZ. Hence, the translated network is also WR and DZ. This now enables the parametrization of the equilibria of the model.

https://doi.org/10.1371/journal.pcbi.1014633.g003

A generalized reaction network consists of two components: the stoichiometric CRN, which contains the complexes obtained after the translation complexes are added to the reactions, and the kinetic-order CRN, which retains the original source complexes of the reaction network, thereby preserving the dynamics of the system. The reactions in red in the kinetic-order CRN are called phantom edges, which connect identical complexes in the stoichiometric CRN. If both stoichiometric CRN and kinetic-order CRN are WR and DZ, then the translated network is WR and DZ. See the Methods section for more information.

Using CRITERIA, the derived equilibrium expression for dCas9 is given by

where k₂₈ and k₂₇ represent the rate constants for the production (R₂₈) and degradation (R₂₇) of dCas9, respectively. This tells us that the equilibrium concentration of the dCas9 species is dependent only on these specific rate constants. So, the equilibrium concentration of dCas9 remains robustly constant for any initial conditions under fixed parameters. This remarkable property is known as absolute concentration robustness (ACR) [19]. This finding has implications for the design of robust synthetic systems:

• Buffering Against Network Fluctuations: ACR suggests an intrinsic buffering mechanism in which the equilibrium concentration of dCas9 remains invariant despite fluctuations in other network components, such as target gene concentrations or guide RNA levels.
• Predictable Tuning: In synthetic biology, achieving predictable and robust behavior is a primary concern. Our discovery shows that the long-term concentration of dCas9 is robustly constant over the concentrations of all other species.
• Simplified Design Parameters: Fine-tuning the long-term concentration of dCas9 in the circuit can only be achieved by altering its production rate k₂₈ or degradation rate k₂₇. This decoupling from other parameters allows for modular circuit design where dCas9 levels can be set independently of the complexity of the broader network.

By maintaining a stable pool of the dCas9 engine, the CRISPRi circuit ensures that the regulation machinery remains consistent in varying cellular environments, effectively anchoring the stability of the toggle switch.

Computational package, CRITERIA

We developed CRITERIA (https://github.com/evvillejo/CRITERIA), a user-friendly, open-source, and publicly available computational package that automates the computation of equilibria of biochemical reaction networks. The framework enhances and extends COMPILES by incorporating several methodological improvements that overcome previously identified limitations (see Fig 4 and its accompanying discussion).

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Fig 4. Side-by-side comparison of the COMPILES and CRITERIA frameworks for deriving analytic equilibria.

For both frameworks, the method starts by decomposing the network into its independent subnetworks, thereby simplifying the analysis by focusing on smaller individual pieces rather than looking at the network as a whole. The next step is to perform network translation for each subnetwork that is not WR and DZ. For COMPILES, translation is performed using the TOWARDZ approach introduced in [23]. On the other hand, CRITERIA uses an EFM-based approach [5]. Both frameworks differ in the last two steps. Once all subnetworks satisfy WR and DZ, COMPILES proceeds by determining the positive equilibria of each subnetwork through the parametrization method in [22]. Finally, the equilibria of the original network are obtained by merging the equilibria of the subnetworks. However, for CRITERIA, it first performs merging of the translated subnetworks into a single network by taking the union of the reactions in the subnetworks before proceeding to compute the positive equilibria of the entire network, which constitutes the last step of the enhanced procedure.

https://doi.org/10.1371/journal.pcbi.1014633.g004

Taking the set of reactions of a CRN as input, CRITERIA outputs the following: an equilibrium parametrization, kinetic deficiency of the GCRN along with the additional equations required by Case 2 of Theorem 2, and the conservation laws of the system. As with COMPILES, CRITERIA does its best to output the most simplified analytic solution in terms of rate constants and free parameters.

Chemical reaction networks

A chemical reaction network (CRN) is defined by three sets [24]. The first set is the set of species , which contains the fundamental units of the network called species. The second set is the set of complexes , which contains entities called complexes that are linear combinations of species. That is,

The coefficients are called stoichiometric coefficients. We use as both the complex itself and the corresponding complex vector . And lastly, the third set is the set of reactions .

The structure of a CRN can be seen as a directed graph where the edges are the reactions, and the vertices are the complexes. So, we can represent reactions as ordered pairs of complexes (i.e., ) or as directed edges (i.e., ). Here, we call the reactant or source complex and the product complex. A weight is associated to each reaction called the reaction rate coefficient. For each reaction , we can associate a reaction vector which serves as the columns of the stoichiometric matrix N of the CRN.

Example 1. Consider the following toy CRN with five reactions:

In this CRN, R₁ represents the conversion of A to C in the presence of B, with B effectively acting as a catalyst for the reaction. The reversible reactions R₂ and R₃ can be interpreted as conversion of A to B, and vice versa. Reaction R₄ models the degradation or removal of C from the system, while R₅ represents the inflow or external production of A. The network consists of three species, six complexes, and five reactions. In particular,

The zero complex is not considered as a species; rather, it is a formal placeholder that represents the external environment of the system, where it acts as a “sink” or “source” of species and complexes. In , A + B acts as the source complex and C + B acts as the product complex. Furthermore, the following are the reaction vectors of the CRN:

These reaction vectors comprise the columns of the stoichiometric matrix, which can be written as

whose rank is given by s = 3.

Since a CRN can be seen as a directed graph, we can draw parallels between concepts in graph theory and CRN theory. In graph theory, we say that two vertices are connected if there is a directed path between them. Furthermore, two vertices are strongly connected if there is a directed path to and from each other. Given these definitions, we call the connected components of the network the linkage classes of the CRN and the strongly connected components the strong linkage classes of the CRN. In other words, a linkage class is a set of complexes that are connected to one another but not to any other complex not in the set, whereas a strong linkage class consists of complexes that are strongly connected to each other. If the number of linkage classes is equal to the number of strong linkage classes, then the CRN is said to be weakly reversible. Alternatively, a CRN is said to be weakly reversible if every reaction is part of a directed cycle.

Example 2. In Example 1, there are two linkage classes, namely {A + B,C + B} and {A,B,C,0}. On the other hand, there are five strong linkage classes which are the following: {A,B}, {C + B}, {A + B}, {C}, {0}. Note that a complex, if not strongly connected to other complex/es, is a trivial strong linkage class. Since the number of linkage classes does not coincide with the number of strong linkage classes, the CRN is not weakly reversible. Alternatively, this can be seen from the fact that reaction R₁ is not contained in any directed cycle, and hence the network fails to be weakly reversible.

One of the central concepts in CRN theory is called the deficiency of the CRN which is denoted by . It is a nonnegative integer with formula , where n is the number of complexes, l is the number of linkage classes of the CRN, and s is the rank of the stoichiometric matrix. It can be interpreted as a measure of the linear dependence among the reactions of the reaction network. The higher the deficiency, the lower the extent of linear independence [20].

Example 3. In Example 1, we have . So, it is a deficiency one network.

A CRN is endowed by kinetics to describe the evolution of the concentration of the species over time. One of the widely-used kinetics is mass-action which states that the rate at which the reaction proceeds is proportional to the product of the concentration of the species in its reactant complex.

Example 4. In our running example, suppose a, b, and c denote the concentrations of species A, B, and C, respectively. Then, the rate function of each reaction under the assumption of mass-action is given by

Together with N, the system of ordinary differential equations for the model is given by

where , , and are the time derivatives of the concentration functions of the species A, B, and C, respectively. The system of ODEs can be written as

The long-term behavior of a system is described by the equilibrium or steady states, which can be solved by setting the time derivatives to zero and then solving for the concentration of the species.

Example 5. The positive equilibrium of the network in Example 1 is given by

where .

Network decomposition

A decomposition of a CRN is induced by a partition of its reaction set [2]. This partition then gives rise to a decomposition of the reaction network into subnetworks. One special type of network decomposition is called independent decomposition, where the rank of the stoichiometric matrix of the whole network is equal to the sum of the ranks of the stoichiometric matrices of its subnetworks [24].

Example 6. In Example 1, the following partition of the reaction set into

induces a decomposition of the CRN into subnetworks and with stoichiometric matrices

respectively. It can easily be verified that s₁ = 2 and s₂ = 1. Since we have equality , the decomposition is independent.

A method was developed in [27] to obtain an independent decomposition of a reaction network that employs a coordinate graph to find the desired decomposition. It was then proven that the obtained independent decomposition in [27] is already the finest (i.e., the independent decomposition with the maximum number of subnetworks) [28].

A key feature of independent decomposition is that it allows us to compute the positive steady states of the entire network by first deriving the positive steady states of each independent subnetwork and then intersecting them. This is formalized in the following theorem which is taken from [24].

Theorem 1. Let be a reaction network with kinetics decomposed into subnetworks and be the restriction of to reactions in . Then

where E is the set of positive steady states of the whole network while is the set of positive steady states of subnetwork . Furthermore, if the network decomposition is independent, then equality holds, i.e.,

Network translation

The parametrization method in [22] requires a CRN to be weakly reversible and deficiency zero. When these conditions are not satisfied, the reaction network must be transformed to obtain a network with the desired structural properties. This is done by the process of network translation, first introduced in [18], which modifies the graphical properties of the CRN but preserves the reaction vectors, and allows the original source complex to determine the kinetics so that the dynamics of the system is preserved.

Network translation can be visualized by the operation of adding or subtracting linear combinations of species, known as translation complexes, from individual reactions [5]. Formally, we let to be a translation complex for reaction k. We represent the operation of translating the reaction by the translation complex as

(8)

for . Here, we add the translation complex to both reactant and product complexes to produce the translated reaction.

Example 7. Going back to Example 1, translating R₄ by and R₅ by gives us the following translated network:

Clearly, the translated network is weakly reversible. Furthermore, it can be verified that its deficiency is equal to zero, which is not the case for the original network.

The translation scheme in (8) produces a new set of complexes, containing and , which are called stoichiometric complexes in the translated network. Since the original source complex determines the kinetics of the translated network, another set of complexes is produced called the kinetic complexes. Therefore, the translated network can be thought of in terms of a generalized chemical reaction network (GCRN).

Generalized chemical reaction networks

In this subsection, we formally define the concept of a GCRN, which was pioneered by Müller and Regensburger [17], and show how this is useful in deriving the analytic equilibria of a reaction network.

Let G=(V,E) be a directed graph with vertex set V and edge set . Furthermore, denote to be the set of all source vertices in G. A GCRN is a directed graph G=(V,E) with two maps

• that assigns to each vertex a stoichiometric complex; and
• that assigns to each vertex a kinetic complex.

Given a CRN with an associated graph G, we can construct a GCRN with graph via network translation such that the resulting system of ODE matches that of the original CRN. We can do this by allowing the kinetic complex to determine the reaction rate while having the stoichiometric complex determine the reaction vector. Doing this for all reactions, we are assured to get the same ODE system since the original reaction rates and stoichiometric matrix are preserved. In such case, the CRN and the GCRN are said to be dynamically equivalent. The two maps y and give rise to two associated CRNs for a GCRN: the stoichiometric CRN and the kinetic-order CRN . Consequently, we can define deficiencies for both the stoichiometric CRN and kinetic-order CRN, which are called effective deficiency and kinetic deficiency , respectively. Both are calculated with the same formula but using the CRN associated with them.

There are two kinds of edges in a GCRN: phantom edge and effective edge. A phantom edge connects identical stoichiometric complexes in the GCRN. Otherwise, it is an effective edge. A phantom edge does not contribute to the system of ODEs since we would get 0 as its corresponding reaction vector. Thus, we assign a dummy reaction rate constant , which is considered as a free parameter, for phantom edges. Lastly, we denote the sets of phantom edges and effective edges of the GCRN by E⁰ and , respectively.

Given these, the main parametrization method is given in the theorem below [2,22].

Theorem 2. Consider a weakly reversible translated network, which is a GCRN. Let be any spanning forest containing all the nodes of the kinetic-order CRN. For each edge of , we define the kinetic difference as the vector produced by subtracting the head kinetic complex by the tail kinetic complex. Furthermore, M is the matrix containing all the kinetic differences as rows where the entries per row are arranged according to the order of species. Let H be a generalized inverse of M (i.e., MHM = M). Finally, define B and C such that im(B) = ker(M) and ker(B) = {0}, and im(C) = ker and ker(C)={0}.

• (Case 1) If the kinetic deficiency is zero, the set of parametrized complex-balanced equilibria is given by
• (Case 2) If the kinetic deficiency is positive, define the equations
  and add them as additional conditions to the parametrization in Case 1.

In this formalization, the product is the Hadamard product (the number of components of is the number of edges in a spanning forest, which can be effective of phantom) with the component of associated with the edge as and tree constant as the sum (over all spanning trees of the kinetic-order CRN towards node i) of the products of the rate constants associated with the edges of each spanning tree. In addition, if the effective deficiency is zero, then the set of positive steady states of the original network is precisely .

Example 8. In Example 1 and its translation scheme in Example 7, we define the GCRN as follows

Here, the complexes in parentheses are the stoichiometric complexes, while the complexes above them are the kinetic complexes. Determining the reaction rates of the GCRN, we have

Moreover, the reaction vectors of the GCRN are the following:

It can be observed that the reaction vectors we have computed coincide with those obtained in Example 1, while the reaction rates agree exactly with those derived in Example 4. Hence, the original CRN and the GCRN in this example are dynamically equivalent. Finally, it can be verified that the GCRN has zero effective deficiency () and zero kinetic deficiency (), so Case 1 of Theorem 2 applies.

Elementary flux modes

Johnston and Burton [5] developed a computational method for performing structural translation through an elementary flux mode-based approach. An elementary flux mode (EFM) represents a minimal, non-decomposable set of reactions that operate at steady state. In other words, EFMs are flux-balanced pathways that cannot be simplified in the sense that it is not possible to remove a subset of active reactions from an EFM and still be able to build a flux-balanced path using only the remaining active reactions [6].

Formally, a vector is an EFM of the CRN if and is not a convex combination of any other . The set is the set of all admissible/feasible flux vectors, sometimes referred to as the elementary flux cone , where is the set of all EFMs of a CRN. We say that is unitary if every entry in every is a one or a zero. Furthermore, covers the reaction set is [5].

If the set of EFMs of a CRN is unitary and covers , then each EFM is completely determined by its support, , which identifies the subset of reactions active in that mode. As a result, it is sufficient to represent an EFM by the positions of its nonzero entries, allowing us to use the binary vector notation (i.e., ) and the corresponding reaction set notation (i.e., ) interchangeably.

The EFMs can be interpreted as the sets of reactions that, if taken in any order, would result in no net gain or loss of any species. In other words, each is a flux path through the reaction network with the property that the net total flux into and out of all nodes/vertices balances. Furthermore, EFMs are nondecomposable flux vectors from which all possible flux vectors of the network can be completely described [6]. That is, the EFMs are the extremal generators of .

Example 9. Note that the stoichiometric matrices of the network in Example 1 and its weakly reversible, deficiency zero translation in Example 7 coincide since both networks have the same set of reaction vectors. Since EFMs are, by definition, elements of the null space of the stoichiometric matrix, the original and translated networks have the same set of EFMs. In particular, they have two EFMs, which are given by .

Reaction-to-reaction graph

A reaction-to-reaction graph is a directed graph whose vertices correspond to the reactions of the CRN [5]. Directed cycles in the reaction-to-reaction graph are formed using the EFMs of the reaction network. Formally, a directed graph is a reaction-to-reaction graph of a CRN if and .

It would be convenient to use the mappings in the subsequent definitions, where and map the source and product of each reaction to its corresponding complex, respectively. That is, we may use the notation instead of . Now, we say that the CRN and the reaction-to-reaction graph are:

• product-to-source compatible (PS-compatible) if, for any .
• common source compatible (CS-compatible) if
• elementary flux mode compatible (EM-compatible) if every EFM of the CRN corresponds to the vertices of a minimal directed cycle in the reaction-to-reaction graph, and vice versa

The theorem below relates the properties of PS-, CS-, and EM-compatibility to a weakly reversible, deficiency zero translation of a network. This is taken from [5].

Theorem 3. Consider a CRN with a set of elementary flux modes which is unitary and covers . If there is a reaction-to-reaction graph which is CS- and EM-compatible to , then there is a CRN which is PS- CS, and EM-compatible with . Furthermore, is a weakly reversible, zero deficiency structural translation of . In particular, the translation complexes required to produce such a translation satisfy the following linear system, which is necessarily consistent:

(9)

Remarks

1. It is important to note that Theorem 3 is only applicable when the set of EFMs of the reaction network is unitary. Consequently, CRITERIA cannot obtain an equilibrium parametrization of a CRN when its set of EFMs is non-unitary. The limitation of the method in [5] is inherited in CRITERIA.
2. As shown in Table 3, REWARDZ without network decomposition was unable to generate a valid translation for Model N because its set of EFMs is non-unitary. However, this limitation was overcome by first performing network decomposition, after which REWARDZ successfully obtained a valid translation.

The goal then is to construct a reaction-to-reaction graph which are CS- and EM-compatible with a given CRN and then enforce PS-compatibility to construct a weakly reversible, deficiency zero translation via solving a linear system.

Example 10. Consider the reaction-to-reaction graph below.

Indeed, the EFMs computed in Example 9 given by correspond to minimal cycles in the reaction-to-reaction graph. Moreover, since there are no reactions with common source complex in the network in Example 1, then the CRN and the reaction-to-reaction graph are automatically CS-compatible. However, the reaction-to-reaction graph is not PS-compatible to the CRN. For instance, consider . Nevertheless, the reaction-to-reaction graph is EM- and CS-compatible with the network in Example 1. Moreover, the set of EFMs . By Theorem, 3, there exists a weakly reversible, deficiency zero translation of the network. Indeed, such a translation was constructed in Example 7.