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. 2023 Jan 11;8(3):3043–3056. doi: 10.1021/acsomega.2c06113

Localization of Noise in Biochemical Networks

Erickson Fajiculay †,‡,§, Chao-Ping Hsu †,‡,∥,⊥,*
PMCID: PMC9878546  PMID: 36713703

Abstract

graphic file with name ao2c06113_0014.jpg

Noise, or uncertainty in biochemical networks, has become an important aspect of many biological problems. Noise can arise and propagate from external factors and probabilistic chemical reactions occurring in small cellular compartments. For species survival, it is important to regulate such uncertainties in executing vital cell functions. Regulated noise can improve adaptability, whereas uncontrolled noise can cause diseases. Simulation can provide a detailed analysis of uncertainties, but parameters such as rate constants and initial conditions are usually unknown. A general understanding of noise dynamics from the perspective of network structure is highly desirable. In this study, we extended the previously developed law of localization for characterizing noise in terms of (co)variances and developed noise localization theory. With linear noise approximation, we can expand a biochemical network into an extended set of differential equations representing a fictitious network for pseudo-components consisting of variances and covariances, together with chemical species. Through localization analysis, perturbation responses at the steady state of pseudo-components can be summarized into a sensitivity matrix that only requires knowledge of network topology. Our work allows identification of buffering structures at the level of species, variances, and covariances and can provide insights into noise flow under non-steady-state conditions in the form of a pseudo-chemical reaction. We tested noise localization in various systems, and here we discuss its implications and potential applications. Results show that this theory is potentially applicable in discriminating models, scanning network topologies with interesting noise behavior, and designing and perturbing networks with the desired response.

I. Introduction

Heterogeneous gene expression and dynamics have been observed in cells with identical genetic backgrounds, cultivated under the same conditions. Such uncertainty, or noise, has been studied in the context of gene regulation and phenotypic variation.1−9 In a (bio)chemical system, especially in cells, component uncertainty can arise at the molecular level from the probabilistic nature of chemical reactions.1 Noise can also arise from fluctuation or uncertainty in external factors, leading to the large cell-to-cell variation observed in the literature.1,10

Regulation of noise is crucial for species survival.11−13 Uncontrolled noise can deteriorate vital functions of cells, especially those that require precision. Noise has also been one aspect of studies of diseases, such as cancer.14−16 On the other hand, regulated noise, such as mechanisms that increase heterogeneity in systems, may be advantageous in coping with stress, diversifying antibodies, increasing functional heterogeneity, etc..17−20 Understanding of the role of noise and its regulation has been growing,21−24 resulting in applications in cancer therapy and drug design.25−27

Theories and computational techniques for modeling intrinsic noise have been developed, but are constantly being modified and improved. The chemical master equation (CME)28 is a stochastic theory that accounts for the production of intrinsic noise in biochemical systems. It is a set of ordinary differential equations (ODE) that describes the state space of a system as a time-dependent probability distribution function.29 For most systems, the CME cannot be solved analytically. An alternative approach is to perform numerical sampling using the stochastic simulation algorithm (SSA).30 SSA is an exact realization of the CME that is very easy to implement, but simulations can be time-consuming, especially for systems of reactions of different time scales. A τ-leaping algorithm that takes large time steps has been introduced, employing statistically estimated means and variances in the reactions. It greatly reduces the computational cost compared to SSA, but it can be less accurate.31 Some implementations that shift between SSA and τ-leaping have been developed to prevent most of the loss of accuracy.31,32 In terms of speed, a less accurate, but faster approach than τ-leaping is the chemical Langevin equation.33−35 With this approach, the error can be large for some systems, especially if the step size is not properly chosen.36,37

The study of biological noise using numerical approaches, like all dynamic simulations, requires data for initial conditions and rate constants, which can sometimes be estimated by fitting.38−42 However, available data are usually insufficient to constrain parameters.43−46 In this case, one possible approach is to randomly draw parameters from a physically meaningful range and to perform simulations for all combinations of those parameters. By so doing, conclusions can be deduced based on the most probable and consistent outcome, assuming that the generally robust nature of biological systems implies considerable independence in system behavior.47−50 This approach is time-consuming and does not necessarily generate correct results. With many possible parameter sets, sensitivity is often studied with statistical quantities only.51−53 In any case, it is highly desirable to obtain properties of a system without knowing the actual parameters.

Linear noise approximation (LNA) has been an important tool in the study of noise, in the form of (co)variances of species in the system.54,55 With a power expansion in the CME, LNA yields a set of ODEs for (co)variances of components. For small systems, LNA can be solved analytically, providing expressions that relate (co)variances to rate constants and initial composition.56 LNA, to some extent, can be used to draw general conclusions, as it has been a powerful technique for deducing various factors in noise propagation.57−60 LNA-derived noise can also be decomposed into noise contributions from individual reactions in the network.61

Another useful mathematical tool in the study of biological dynamics is the law of localization.62,63 Perturbations of reaction rates, such as inhibition, knockdown, or upregulation of genes or enzymes, could promote changes in steady-state composition that can be monitored experimentally64 and can be predicted through localization analysis. Based only on topology, a sensitivity matrix formulated by inverting the Jacobian of fluxes with appended null space of the stochiometric matrix, the law of localization predicts the overall perturbation response of a network. This theory gives rise to the concept of buffering structures, which are subnetworks of a network such that perturbation of reactions inside a buffering structure only affects species inside, while the rest of the network is unaffected. Such phenomena can be associated with systems where important compounds are under tight homeostasis and are robust to perturbations.65 It was also reported that a large number of species in a carbon metabolic network do not respond to multiple enzyme knockouts,64 which can be attributed to buffering structures. Buffering structures have also been applied in the study of bifurcations.66,67 For small systems, buffering structures can be easily identified using simple rules.63 However, the study of large systems may still rely on numerical computations, which require parameter values that can be scanned or sampled. With the law of localization, there is no need to perturb each parameter and to solve for steady-state ODE of the system repeatedly. This makes the law of localization a very efficient method for finding perturbation responses. The growing availability of multiple high-throughput (knockdown/knockout) data68 and (cellular) perturbation screens69 that can be compared with predicted response patterns from the law of localization may promote better network reconstruction,70−77 model discrimination,78,79 and network engineering.80

Since LNA generates a set of ODEs for (co)variances, it would be interesting to see if such ODEs can be further analyzed using the law of localization. By so doing, i.e., combining LNA with the law of localization, it becomes possible to deduce the noise response or relative change in (co)variance after perturbation of a network of reactions. In the present work, we focus on noise predicted by the law of localization in open systems. We limit our scope to open systems that allow exchange of material with the surrounding environment. These typically have at least an incoming and an outgoing flux with the surroundings. This type of system is commonly found in nature, but because the full scale of natural systems is undetermined, we can only study a portion of such systems, in which at least one component is produced via inflow, and at least one component is degraded via outflow. Systems that do not exchange molecules with their environments are sometimes called closed systems,81 and they have a different set of constraints; thus, they require a separate set of considerations, which we plan to investigate in future work.

Here, we address the study of noise55 by inferring noise dynamics based on network structure. We adopted the law of localization in a chemical reaction network62 and combined it with LNA54,82 to derive a mathematical approach that can help to infer the behavior of noise. In doing so, we can deduce a pseudo-chemical reaction that includes (co)variances. Together with a set of chemical reactions, our results provide a qualitative and quantitative picture of noise dynamics, from a simple look-up to numerical calculations. We tested our approach on selected systems, and we provide a rationale for its implications and potential applications.

II. Theory and Methods

We first briefly introduce LNA and network localization in this section. Noise localization is developed by combining the two, which we describe at the end of this section.

II.I. Linear Noise Approximation

With a power expansion of the system size in the CME,82 a linear Fokker–Planck equation83 can be obtained, as shown in eqs 1–3

II.I. 1

where

II.I. 2

and

II.I. 3

In the equations above, x is a vector of length N describing the amounts of various chemical species in the system, ξ is a random variable equivalent to x – ⟨x⟩, w is the propensity vector, which is a function of x, and the vector of rate constant k, t is the time, V is the stoichiometric matrix, and Π is a probability distribution of ξ. Assuming that Π follows a Gaussian distribution, eq 1 can be simplified to

II.I. 4

where C is the covariance matrix of x.

At steady state, eq 4 vanishes to zero.54,84 For simple systems, an analytical solution to the covariance can be found, but a numerical approach is still needed for large systems.

II.II. Law of Localization in Chemical Networks

The law of localization in chemical networks62 takes advantage of the null space of the stoichiometric matrix, which constrains changes in reaction fluxes. A set of M chemical reactions among N species can be represented as a product of a stoichiometric matrix and propensities, or fluxes, as shown in eq 5

II.II. 5

The flux vector, w(k,x), is a function of the component vector x, and rate constants vector k, with wj corresponding to the flux of reaction j. At steady state, eq 5 vanishes to zero. Perturbation of the jth reaction by a change in the rate constant kj* at steady state results in perturbation of w, which is mathematically equivalent to

II.II. 6

In considering the influence of a perturbation on reaction rates to changes in the steady states, one can consider

II.II. 7

Following ref (62), a null space could exist, which is spanned by vectors cz orthogonal to the raw vectors of V, with the combination of reaction fluxes that is not constrained by the steady-state condition. Therefore, the new steady-state condition with a change in reaction rate can be written as

II.II. 8

where uz denotes coefficients for the null space. Rearrangement and transposition of terms in eq 8 followed by factorization into matrix products gives us the following expression

II.II. 9

where

II.II. 10

and

II.II. 11

In these expressions, Z is the dimension of the null space of V; S is the network sensitivity, composed of δx, which is (∂x/∂k)δk, and coefficients of the null space of V, the row vector uz, respectively; A is a matrix of the derivative of w with respect to x, with null space of V appended; and ej is (∂wj/∂kj*)δkj*. With these manipulations, the sensitivity S yields information for network localization.

For example, in the case of knockdown or inhibition, δkj* is negative, and ej can be assumed to be −1, making S equal to the inverse of A. The network sensitivity S is a table of responses in which columns are reactions and the first N rows are chemical species. Entries in the table are either zero or simple mathematical expressions. A zero entry indicates that perturbing a particular reaction does not change the corresponding chemical species at steady state. For nonzero responses, when kj* is inhibited, the sign of the sensitivity indicates the corresponding direction of changes in the species concentration. Contrarily, when kj* is upregulated, ej is positive in eq 9 and the fate of species upon reaction perturbation follows the same principle. The main implication of the law of localization is that response patterns exhibit localization and hierarchy. This means that perturbation of an upstream reaction does not necessarily affect downstream reactions. Buffering structures, in which perturbations can be confined, can be summarized as a metabolite and reaction subset pair, which are components with nonzero responses in a column of S and their associated reactions.

II.III. Noise Localization

We further analyze noise55 with law of localization theory62 for open systems, where nontrivial results of perturbation in the noise can be formulated. To formulate a network sensitivity matrix that includes noise as (co)variances for open systems at steady state, the component vector x and unique nonzero elements of the covariance matrix C are combined into vector G, shown in eq 12.

II.III. 12

The differential equation of G is a column vector representing unique expressions of the concatenation of eqs 4 and 5 as follows

II.III. 13

Expansion of eqs 4 to 5 and substitution into eq 13 gives the expression

II.III. 14

where g’s are defined as

II.III. 15
II.III. 16
II.III. 17

and

II.III. 18

Noting that w is a function of x and k, we can simplify the expression to a linear combination of w and w′C, where w′ is the derivative of w with respect to x.

II.III. 19

Equation 19 can be simplified into a form similar to eq 5 as follows

II.III. 20

and

II.III. 21

and

II.III. 22

where Q and z are a pseudo-stoichiometric matrix and a pseudo-propensity or a pseudo-flux vector, respectively. Entries in the columns for a row of Q can be repeated if there is more than one nonzero element in the summation over index k in eq 19. When i = j, similar terms will appear for a particular l, k value giving rise to a factor 2 in some terms. Therefore, the η in eq 21 is 2 when i = j in eq 19, otherwise η = 1. The pseudo-propensity vector z in eqs 20 and 22 is composed of functions of k, x, and C. Since eq 20 resembles a regular differential equation for chemical reactions, we can deduce a pseudo-chemical reaction from eq 20 that includes the covariance as species. This pseudo-chemical reaction describes the flow of noise in the system from the source to the reservoir. An algorithm that extracts the pseudo-chemical reaction and factorizes a system of ODEs into a form similar to eq 20 is provided in the following link; “https://github.com/efajiculay/NoiseLocalization.”

Following this derivation and using the law of localization, we derive a pseudo-network sensitivity matrix Inline graphic similar to eq 9 that includes the noise terms as species. This time, the effect of perturbation on the (co)variance is also explained. Localization and perturbation hierarchies with both chemical species and (co)variance terms can be deduced from Inline graphic. We can also adopt the metabolite subset pair from ref (62) to identify buffering structures in the pseudo-chemical reaction network and can infer bifurcation in noise similar to ref (66).

From eqs 9 and 11, the corresponding A and S matrix for noise localization is as follows

II.III. 23

and

II.III. 24

. As in the network localization, c′ is the null space of the new stoichiometric matrix Q, and u is now composed of row vectors for the coefficient of the null space. The pseudo-sensitivity matrix can be expressed as

II.III. 25

where e′j is negative (−1) for inhibition and positive (+1) for activation or upregulation. Noise localization is similar to the law of localization, and the response (increase, decrease, or unchanged) can be predicted from the sensitivity matrix, except for cases in which the sign of the expression can be both negative and positive.

Equation 25 offers a way to perform noise localization analysis, which provides information on the perturbation response of system (co)variances. However, multiple columns in Inline graphic can be associated with the same rate constant. This problem can be resolved by a linear combination of columns controlled by the same parameter. Column labels in Inline graphic are pseudo-propensities and row labels are pseudo-components. A particular rate constant kj can be present in more than one column label. For a pseudo-component in each row, the effective response for a perturbation of kj is the sum of all columns in that row where kj appears in the column label multiplied by the steady-state value of the column label, as shown in the following equation

II.III. 26

This scaling or weighting is necessary for numeric noise localization to match the linear noise approximation (LNA)-independent perturbation responses. The pseudo-propensities z serve as the probability or weight of the response.

II.III.I. Variance Response

With eqs 23–26 we can analyze the localization structure of variances and covariances. However, we find that it is helpful to summarize the variance first since there are cases in which we can infer the variance via the law of localization.

Under mass-action kinetics, linear open systems consist of two types of elementary steps, a zero-order production reaction, and a first-order degradation or transformation reaction. Knowing that z is a function of w and w′C, the term ∂z/∂G from eq 23 can provide the behavior of species and variances. For first-order reactions, w is composed of linear products of k and x, making w′ a function of k only. This makes ∂z/∂G for the variance Ci,i and the species xi equal or ∂w/∂xi = ∂w′C/∂Ci,i = w′i. Upon inversion of the Inline graphic matrix, which is a sparse matrix, these two derivatives give the same result, which indicates that the response of the variance Ci,i, and the species xi is the same. For zero-order reactions, w is only a function of k, and such terms in w′ equal zero. This means that ∂z/∂G is zero for variances and species making them behave in the same way. Therefore, when the form of w is either zero (independent of x) or first order (monomolecular) with respect to xi, the following relation holds

II.III.I. 27

For systems in which eq 27 holds, the response of Ci,i with respect to reaction perturbation follows that of xi. In this case, the response of the variance can be deduced directly from the law of localization following the sign of the sensitivity matrix in eq 9.

If the form of w is complex with respect to xi, a noise localization sensitivity matrix is required to describe the fate of Ci,i since there is no assurance that eq 27 still holds. For example, w can include Hill-type expressions and other nonlinear functions arising from multimolecular reactions. In this case, rate constant perturbation can be inferred from eq 25.

In an open system, concentration perturbation generally does not affect the variance. However, for nonlinear systems that exhibit multiple stable fixed points, concentration perturbation might allow shifting between fixed points giving complicated perturbation responses. This, however, is not currently accounted for.

The response of variance can also be used to estimate the most likely response of the coefficient of variation (CV) defined as Inline graphic. As long as var(xi) is less than ⟨xi⟩2, CV will decrease as ⟨xi⟩ increases. For Poisson processes, var(xi) = ⟨xi⟩ and deviation from Poisson is very unlikely to result in var(xi) ≥ ⟨xi⟩2. In general, the fano factor or var(xi)/⟨xi⟩ is a small number. This set of statements suggests that if var(xi) and ⟨xi⟩ increase, then CV(xi) most likely decreased. If var(xi) and ⟨xi⟩ decreases CV(xi) most likely increased. Under certain conditions, for nonlinear cases, opposite responses between var(xi) and ⟨xi⟩ might be observed. If var(xi) decreased and ⟨xi⟩ increased, CV(xi) definitely decreased. If var(xi) increased and ⟨xi⟩ decreased, CV(xi) certainly increased. For instance, where either var(xi) or ⟨xi⟩ has no response, CV(xi) can still be logically predicted since CV(xi) is directly proportional to Inline graphic and inversely proportional to ⟨xi⟩. Finally, if both var(xi) and xi have no response, CV(xi) surely has no response.

II.III.II. Covariance Response

Since eq 25 involves the inverse of a large matrix, it is cost-effective to take advantage of cases in which the steady-state value of the covariance is zero. In open systems, the covariance Ci,j≠i is zero at steady state for zero-order and first-order (monomolecular) reactions in which reactants or products are no more than one species. This was derived in ref (81) through CME, in which the probability of species becomes a product of a Poisson distribution at steady state. The mean and variance of a Poisson random variable are equal and the covariance between independent Poisson random variables is zero. However, the derivation in ref (81) does not account for monomolecular systems that include autocatalysis and splitting of reactants, i.e., production of more than one product in a monomolecular reaction. For such systems, if the sum of the rows in each column in eq 2 is either zero or negative, then all covariances are zero. Otherwise, there is no assurance that all covariances are zero. The simple gene regulation model in Figure 1c demonstrated in Section III.I is one such exception. It is a linear system with nonzero covariance.

Figure 1.

Figure 1

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Monomolecular open systems analyzed using noise localization. Reactions are labeled with numbers and components with letters. (a) Two-state reversible reaction in a linear chain. (b) This example is a four-state reaction with a cycle constructed based on ref (62). (c) Simple gene regulation model.

Here we summarize our current understanding of cases with nonzero covariance:

  • 1.

    Ci,j≠i is nonzero for systems in which the total number of molecules is conserved. This is because a mathematical relationship between any two species can be defined by exploiting the mass–balance relationship.

  • 2.

    Ci,j≠i can be nonzero if xi and xj are co-produced or co-consumed either directly or indirectly. This is usually observed in multimolecular reactions, disintegration reactions, breakdown or dissociation reactions, etc.

  • 3.

    Ci,j≠i can be nonzero if the system is described by a complicated propensity function containing functions other than constants and first-order terms.

For nonzero covariance, the ODE should be added to the list of ODEs in eq 13 to reflect their perturbation behavior in the pseudo-sensitivity matrix.

III. Results and Discussion

In the following section, perturbation analysis using noise localization is applied to selected systems and compared to LNA with rate constant perturbations or to direct evaluation of the analytical expression whenever applicable. The result is tabulated where dynamics of noise upon perturbations are easily seen. All perturbations applied in this work are inhibitions. The effect of upregulation can be deduced by flipping the signs observed from the response obtained. Section III.I focuses on monomolecular systems in which all mass-action kinetics are either zero or first-order reactions. Discussions of multimolecular systems follow in Section III.II. Section III.III is an application of noise localization to the glycolysis and TCA cycle of Escherichia coli. Codes and derivation for each example can be found at https://github.com/efajiculay/NoiseLocalization.

III.I. Monomolecular Systems

Figure 1 shows some examples of monomolecular systems. These types of systems are generally linear, involve zero and first-order reactions, and may have an analytical solution for the network sensitivity matrix under noise localization.

The two-state reversible reaction shown in Figure 1a is a typical monomolecular open system. Perturbation of the initial composition and reaction rates for this system are summarized in Table 1. The initial amount of each component does not affect the magnitude of steady-state composition and noise. The covariance between A and B is zero in all conditions, as the reactions have at most one species of reactant or product, as discussed in the previous section. If we decompose the noise using the method described in ref (61), reactions 2 and 3 contribute negatively to the covariance, but reactions 1 and 4 cancel those contributions. This is because the total number of species at steady state is constantly disturbed by the firing of all reactions, making it a random variable and destroying coupling between species. The same zero-covariance also holds in the system depicted in Figure 1b.

Table 1. Perturbation Effect in a Two-State, Reversible, Open Systema.

    parameters
   LNA estimate (rounded)
row no. case A0 B0 k1 k2 k3 k4 A B CAA CAB CBB
1 original 100 10 200 50 55 20 15 10 15 0 10
2 +A0 500 10 200 50 55 20 15 10 15 0 10
3 +B0 100 200 200 50 55 20 15 10 15 0 10
4 +k1 100 10 250 50 55 20 18.75 12.5 18.75 0 12.5
5 +k2 100 10 200 100 55 20 7.5 10 7.5 0 10
6 +k3 100 10 200 50 90 20 22 10 22 0 10
7 +k4 100 10 200 50 55 200 5 1 5 0 1
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a

+ indicates an increase in value compared to row 1.

Figure 2 includes a summary of the resulting sensitivity matrix for systems in Figure 1a,b. The perturbation effect on these components matches the variance since those systems are monomolecular. The sensitivity matrix is also sparse. For example, in (a), inhibition of k2 or k3 only affects A and CAA. In (b), inhibition of k2 affects A and CAA, k3 affects B and CBB, and k4 affects A, B, D, CAA, CBB, and CDD. Under the chosen inhibition of the rate constant in each row, buffering structure can be obtained with the pseudo-components that respond in each column, as in the law of localization.62 These structures confine the effects of perturbation, for both components and variances. In this case, the buffering structure is a consequence of the fact that at steady state, total inflow is equal to total outflow. With a perturbed rate constant, the system regains flux balance via increased or decreased quantities of nearby components, which is also reflected in the noise.

Figure 2.

Figure 2

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Summary of the perturbation response for the monomolecular open systems in Figure 1a,b using symbolic noise localization. Pink denotes a reduction in magnitude. White indicates no effect, and blue represents an increase in magnitude. Ci,i corresponds to the variance of species i, and ki’s are rate constants.

The simple gene regulation model shown in Figure 1c is special because it is a lumped reaction. The actual mechanism would involve many elementary steps, some of which are multimolecular reactions. In reaction 3, where m appears to form P and m, it is not m that forms P, but m serves as a pattern or template in the synthesis of P. The sum of the columns of eq 2 for this system includes positive terms, which imply that some covariances may not be zero.

Figure 3 is a summary of the distribution of responses for Figure 1c, using numeric noise localization. Responses of the variance match those of corresponding species and covariances have a nonzero response with the same direction of changes in each row as P. We note that Cmp is not zero since production of P is correlated with m in this model. In this system, the exact direction of the response can be predicted with a high level of confidence.

Figure 3.

Figure 3

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Perturbation response for the simple gene regulation model in Figure 1c using numeric noise localization summarized for 100 samples. Pink stands for a reduction in magnitude. White indicates no effect, and blue represents an increase in magnitude. Ci,i corresponds to the variance of species i. Ci,j corresponds to the covariance between species i and j. km and rm are the production and degradation rate of the mRNA m, kp is the rate of translation from mRNA to protein P, and rp is the degradation rate of P. The rate constants and initial composition were sampled uniformly, between the ranges of 80–200 and 0–10, respectively. Responses between ±1.0 × 10–10 are regarded as zero.

III.II. Multimolecular Systems

Multimolecular systems are generally nonlinear, making noise response inference complicated, and often analytically intractable. One simplification is to omit the zero-covariance term in eq 13, as discussed in the previous section. However, for multimolecular systems, it is difficult to judge whether a covariance term is zero. A solution to this is to sample random parameters and perform numeric noise localization. In this case, a range −v to +v needs to be defined, in which v is a small threshold within a number to be considered zero. Responses from eq 25 need to be rescaled with the columns or pseudo-propensities that serve as weights of the responses. This is to reduce the effect of nonzero responses associated with very low fluxes. Responses associated with very low steady-state pseudo-component values in each row can be considered as zero, or that row can be omitted. This is because even if they do respond to perturbation, the effect is negligible and indistinguishable from no effect. The detail of the rescaling is reflected in eq 26 and mentioned in step 10 of the steps in noise localization in the Supporting Information.

The existence of multiple fixed points is also likely to cause problems by introducing more uncertainty into predicted noise dynamics upon perturbation. Each fixed point may have different noise dynamic behavior and some random initial conditions or small perturbations may switch between any stable, fixed points.

This subsection focuses on the use of noise localization in analysis of multimolecular open systems (Figure 4). The first two systems differ only slightly in topology and the last system is an example from the law of localization.62 Those systems are chosen to compare different complexities of bimolecular open systems. The system in Figure 4a focuses on a simple bimolecular reaction. Figure 4b captures the effect of an upstream bimolecular reaction on a downstream monomolecular system. Finally, Figure 4c shows the effect of a bimolecular reaction on upstream monomolecular reactions.

Figure 4.

Figure 4

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Hypothetical examples of bimolecular open systems. (a) Bimolecular reaction in which both components experience production and degradation, and binding of the two molecules leads to removal. (b) Here, an additional monomolecular reaction is included in the system depicted in (a). (c) Complicated monomolecular reaction with a bimolecular reaction at the end, constructed based on ref (62).

The system shown in Figure 4a is small enough to calculate steady-state analytical expressions for pseudo-components. In eqs 28–30, the steady-state expression for L, R, and CLR is provided. The noise expression for CLL and CRR is highly dependent on L and R, respectively, but is not shown.

III.II. 28
III.II. 29
III.II. 30

From the expressions above, it is clear that pseudo-components are highly dependent on parameters and are expected to respond to perturbation of any parameters. Equation 28 suggests two possible solutions that are expected from the quadratic term introduced into the propensity by the bimolecular elementary step. In this system, one of the solutions gives a positive L, which is the physically meaningful solution.

The most probable perturbation response of the system, depicted in Figure 4a, is summarized in Figure 5 at two zero thresholds, ±1.0 × 10–3/2 and ±1.0 × 10–10. The appearance of a white response in (a) is due to the loose cutoff, which suggests that the choice of response cutoff range is crucial. There is no white response in (b), which is correct, based on eqs 28–30. Therefore, we believe that the cutoff chosen in (b) is strict enough for numeric tests.

Figure 5.

Figure 5

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Perturbation responses for multimolecular open systems in Figure 4a using numeric noise localization, summarized for 100 samples. The rate constants and initial composition were uniformly sampled between the ranges of 80–200 and 0–10, respectively. Pink stands for a reduction in magnitude. White indicates no effect, and blue represents an increase in magnitude. Ci,i corresponds to the variance of species i. Ci,j corresponds to the covariance between species i and j. ki’s are rate constants. (a) Responses between ±1.0 × 10–3/2 are regarded as zero. (b) Responses between ±1.0 × 10–10 are regarded as zero.

Figure 6 summarizes the distribution of response patterns for the system depicted in Figure 4b. Response patterns of the variances CRR, CLL, CAA, and CBB, are matched to those of species, R, L, A, and B, except under perturbation from k6, in which the responses of CBB and B are different. Species B has zero response, which can be confirmed by the steady-state solution that is independent of k6. However, interestingly, upon inhibition of k6, the noise strength on B, CBB, increases. In other words, with inhibition of k6, the mean concentration of B remains the same, but the dispersion is increased. We have confirmed the positive response using numeric and semianalytical LNA, and this is a rather nonintuitive outcome from a system that contains a bimolecular reaction. Noise localization can be used in the search or screening of such behaviors, which can be utilized in the design of networks with robust steady-state composition and desired noise behavior.

Figure 6.

Figure 6

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Perturbation dynamics for the multimolecular open systems in Figure 4b using numeric noise localization summarized for 100 samples. The rate constants and initial composition were uniformly sampled between the ranges of 80–200 and 0–10, respectively and responses between ±1.0 × 10–10 are regarded as zero. Pink stands for a reduction in magnitude. White indicates no effect, and blue represents an increase in magnitude. Ci,i corresponds to the variance of species i. Ci,j corresponds to the covariance between species i and j. ki’s are rate constants.

Species A and B form a buffering structure downstream of the bimolecular reaction (Figure 6), in which perturbation of k6 and k7 yields results expected from the law of localization. Some of the covariances appear to be a mixing of responses of the species involved. For instance, CRA under k4 perturbation has a nonzero distribution that appears separately in either R and A alone. Most covariances have nonzero responses, which indicates that the presence of a bimolecular step couples most components. The bimolecular reaction k5 also affects all components and their (co)variances, including upstream and downstream components.

The system shown in Figure 4c is composed of an upstream monomolecular structure with nested topologies, a high degree of branching, and a downstream bimolecular reaction. This system is expected to show complex behavior. Figure 7 shows the most probable outcome for the perturbation of reactions in Figure 4c using numeric noise localization. We have verified that the patterns shown are in good agreement with independent LNA simulations. The result is consistent with the law of localization for species A, B, C, D, E, F, G, H, I, and J. Perturbation of k3 and k4 is confined to A and B, which are monomolecular buffering structures and did not affect any pseudo-components except their substrate, CAA and A for k3 and CBB and B for k4. Perturbation of k14 and k15 also shows significant buffering since their substrates are also monomolecular buffering species. The perturbation response of CA,j (omitted in the figure) and CB,j (omitted in the figure) where j is any of the species, matches what is expected from Poisson behavior in which covariances are zero and variances match the species response. Other noise terms show greater deviation from Poisson behavior, but follow Poisson behavior in at least one of the perturbations.

Figure 7.

Figure 7

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Most probable perturbation response for multimolecular open systems in Figure 4c using numeric noise localization summarized for 100 samples. Pseudo-components with no response in any perturbation are removed. The rate constants and initial composition were uniformly sampled between the ranges of 80–200 and 0–10, respectively, and responses between ±1.0 × 10–10 are regarded as zero. Pink stands for a reduction in magnitude. White indicates no effect, and blue represents increases in magnitude. Ci,i corresponds to the variance of species i. Ci,j corresponds to the covariance between species i and j. ki’s are rate constants.

In Figure 7, there are several interesting cases of patterns of pseudo-components. For example, if we compare species C and the corresponding variance CCC, C does not respond to inhibition of k6, k7, k9, k11, and k12, but CCC has a nonzero response. Similar cases are also observed for other pairs of pseudo-components such as D and CDD, F and CFF, etc. Since the bimolecular reaction is downstream of C, D, and F, these differences in perturbation behavior suggest that a downstream multimolecular reaction can alter the noise response of upstream components.

III.III. Glycolysis and TCA Cycle

In this subsection, noise localization is used in modeling the perturbation response to the glycolysis and TCA cycle of E. coli.62,63,85 The network consists of 28 species and 46 reactions giving 406 unique (co)variances with a total of 434 pseudo-components. Figure 8 shows a graphical representation of the network in the left panel and associated reactions in the right panel. This system involves monomolecular, bimolecular, reversible reactions and nested topologies. The full name of the species in the network is provided in Section S3 in the Supporting Information.

Figure 8.

Figure 8

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Glycolysis and TCA cycle of E. coli. This network is constructed based on refs (62, 63), and (85) and the associated reactions are written in BioSANS56 software topology format. Exact names of each species in the network are provided in Section S3 in the Supporting Information. ki’s are rate constants with the subscript a or b corresponding to forward and backward reactions for reversible steps.

Figure 9 summarizes the response pattern for the glycolysis and TCA cycle of E. coli using (a) Noise localization and (b) LNA with parameter perturbations. Response patterns are in good agreement with an overall match of 98.4%, based on tile colors. The discrepancy is mainly because noise localization is under the assumption of infinitesimal perturbation, whereas with parameter perturbation, LNA uses actual perturbation of each parameter. As the perturbation in LNA approaches zero, the result will be closer to what we can get from noise localization. This is a limitation of noise localization in a realistic scenario in which the actual perturbation is unknown and can be large. However, noise response predicted using noise localization is still quite good under large perturbation, as can be seen in Figure 10. As the strength of inhibition increases from right to left along the x-axis (reverse direction), the % match between the responses of noise localization and LNA with parameter perturbation drops, but still above 90%. Moreover, noise localization is faster compared to LNA with parameter perturbation, especially for very large systems. In this system, noise localization is twice as fast as LNA with parameter perturbation when both methods are using multiprocessing. The difference in speed is expected to grow exponentially with system size and as the number of cores for parallel runs becomes exhausted.

Figure 9.

Figure 9

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Most probable perturbation response of the glycolysis and TCA cycle of E. coli summarized for 100 samples. The rate constants and initial composition were uniformly sampled between the ranges of 80–200 and 0–10, respectively. Pink stands for a reduction in magnitude. White indicates no effect, and blue represents an increase in magnitude. Responses between ±1.0 × 10–10 are regarded as zero. Ci,i corresponds to the variance of species i. Ci,j corresponds to the covariance between species i and j. ki’s are rate constants. (a) Result of noise localization. (b) Result using LNA with individual parameters perturbed.

Figure 10.

Figure 10

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Effect of perturbation strength on the accuracy of predicted responses. k_original and k_perturb are the rate constants before and after perturbations, respectively; % match is the ratio of the number of responses that noise localization correctly predicts compared to responses predicted using LNA with parameter perturbations.

Figure 11 shows the most probable perturbation response of the metabolites in Figure 8 under the perturbation of rate constants in each column. This response pattern matches what can be predicted using the law of localization. The corresponding variance response is shown in Figure 12. Even if the metabolite response is sparse, the covariance response is not. For nonlinear systems such as the glycolysis and TCA cycle of E. coli, the variance does not always respond the same way as the species. This result indicates that inclusion of the (co)variance response provides more discrimination power when evaluating network topologies.

Figure 11.

Figure 11

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Most probable perturbation response patterns for metabolites in the glycolysis and TCA cycle of E. coli. This result is a portion of that shown in Figure 9a. The columns are the rate constants with kf and kb for forward and backward reactions, respectively, in the same order as in Figure 8, and the rows are the metabolites.

Figure 12.

Figure 12

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Most probable perturbation response patterns for the variance of the metabolites in the glycolysis and TCA cycle of E. coli. This result is a portion of that shown in Figure 9a. The columns are the rate constants with kf and kb for forward and backward reactions, respectively, in the same order as in Figure 8, and the rows are the metabolites.

Numeric noise localization provides a way to obtain the most probable response to reaction perturbation without the need to perturb each parameter. It is an efficient way to analyze a network, especially compared to numeric propagations such as SSA and SSA-derived algorithms for perturbation behavior. It is also better than performing LNA with N parameter perturbation, which entails performing N + 1 separate LNAs per sampled parameter.

Being able to quickly obtain the response patterns of both species concentrations and their associated noise is beneficial for inferring the reaction network of a system, a highly desirable, but difficult task. Ideally, a good model should have predicted dynamics, together with a perturbation profile that is highly consistent with experimental results. With our approach, a fast test over many different possible network structures becomes much more feasible. Noise/uncertainty can also be derived from single-cell assays, reporter-based methods, flow cytometry experiments, etc. If these were obtained by perturbing one or several components that affect the flux or rates, our approach offers more insight from the data by testing a great number of possible networks. With modern databases such as Library of Integrated Network-Based Cellular Signatures (LINCS69), KnockTF,68 Human Metabolome Database,86 etc., reverse engineering or inference of the true network and pathways will eventually become feasible, and we expect that inference from noise behavior, such as our present work, offers additional clues and constraints in the process.

IV. Conclusions

In this work, we show how qualitative and quantitative noise dynamics can be inferred by exploiting LNA and the law of localization together. The noise localization scheme we developed relies on the sensitivity matrix, which summarizes perturbation without the need to perturb parameters. It provides a way to capture the response in noise, under a perturbation of reaction rates, which can be very efficient compared to traditional approaches that require both parameter sampling and numerical perturbation.

Using the method presented here, variances and covariances in both monomolecular and multimolecular systems can be studied. Monomolecular open systems generally provide concrete answers regarding the fate of pseudo-components in response to perturbation. On the other hand, multimolecular open systems generally show greater uncertainty in the fate of pseudo-components (especially the covariance), subject to perturbation. Nevertheless, our noise localization theory can be useful to discriminate different models compared to experimental data, to help researchers decide how to perturb a given system based on network structure, to render a desired change of state in noise or chemical composition, and to scan interesting noise behavior from a given topological network.

Acknowledgments

The authors gratefully acknowledge support from the National Science and Technology Council (Project 111-2123-M-001-003) and Academia Sinica. They thank Dr. Ching-Cher Sanders Yan and Prof. Je-Chiang Tsai for their helpful discussions.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.2c06113.

  • Steps of noise localization, example of noise localization, and names of species in the carbon metabolic TCA cycle of E. coli (PDF)

The authors declare no competing financial interest.

Supplementary Material

ao2c06113_si_001.pdf (118.1KB, pdf)

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