The algorithm deployed by “ReDirection” is mathematically rigorous and biochemically relevant, and has been extensively discussed (Kundu, 2023a; Kundu, 2023b). Briefly, a biochemical network is modeled as the sparse stoichiometry number matrix p ∼ S p ⊂ Z J × I ═ and is a collection of i -indexed i = 1,2 … I ═ r -reaction vectors r i ∈ Z j (Def. (1a)) (Kundu, 2023a; Kundu, 2023b). Each reaction vector is populated by combinations of j -indexed j = 1,2 … J m -stoichiometry numbers m j ∈ Z of J -reactants/products (Def. (1b)) (Kundu, 2023a; Kundu, 2023b). The modeled biochemical network is subsumed to operate under several biochemically relevant numerical constraints. These include lower bounds for the numbers of reactants/products and reaction vectors, modeling a reaction vector as the interaction between one or more pairs of molecules (enzymatic or non-enzymatic) with differently signed stoichiometry numbers, equilibrium S ≃ 0 , and fixed outcomes for a participating reaction (forward, reverse, and equivalent) (Def. (2)) (Kundu, 2023a; Kundu, 2023b).

“ReDirection” is assessed by the time needed T   min to unambiguously assign an outcome to every reaction (Def. (3)). This depends on the architecture and complexity of the numerical values that constitute the stoichiometric number matrix of the modeled biochemical network, the nullity of the null space, and a network-suitable null space-generated subspace (Kundu, 2023a; Kundu, 2023b). For a stoichiometry number matrix, the desired null space is v k ∈ V ⊂ R I ═ | v k T × S p = 0 → T , (Def. 4) where V ≝ N u l l   s p a c e   o f   S p , (1) # V ≥ 2 , (2) k = 1,2 … K , (3) K = # V , (4) S p ⊂ Z J × I ═ . (5) “ReDirection” combinatorially sums the vectors of the null space and, thence, each null space-generated subspace (Kundu, 2023a). This results in several subsets of vectors which contribute to the cardinality of each null space-generated subspace and may be summarized. We describe this comprehensive null space-generated subspace as the set which contains trivial vectors, along with redundant or finite occurrences of non-trivial and identical null space vectors (Kundu, 2023a): V ═ = V ∪ H ═ ∪ H ∪ L , (Def. 5) where V ≝ N u l l   s p a c e   o f   u s e r − d e f i n e d   s t i c h i o m e t r y   n u m b e r   m a t r i x , L ≝ S e t   o f   t r i v i a l   n u l l   s p a c e   v e c t o r s , (Def. 6) H ≝ S e t   o f   u n i q u e   n u l l   s p a c e   v e c t o r s , (Def. 7) H ═ ≝ S e t   o f   v e c t o r s   w h i c h   h a v e   o n e   o r   m o r e   s u b s e t s   o f   i d e n t i c a l   v e c t o r s . (Def. 8)

Rewriting Def. (4) to include these vectors yields v k ∈ V ═ ⊆ V ⊂ R I ═ | v k T × S p = 0 → T , (Def. 9) where V ═ ≝ C o m p r e h e n s i v e   s u b s p a c e   o f   V   w i t h   c a r d i n a l i t y   # V ═ , V ≝ N u l l   s p a c e   o f   S p , v k = 0 →   T r i v i a l   n u l l   s p a c e   v e c t o r , (6) v k ≠ 0 →   N o n − t r i v i a l   n u l l   s p a c e   v e c t o r , (7) k = 1,2 … K , K = # V . ═

We now enumerate various cases that may arise when we combinatorially sum non-trivial vectors: C a s e   1 : I f v 1 ≠ v 2 … ≠ v K ≠ 0 , t h e n , v 1 , v 2 … , v K ∈ H ⊂ V ═  and  H ═ = ∅ , Def . 10 ⇒ # H = K ; # H ═ = 0 . 8 , 9 C a s e   2 a : L e t v 1 = v 2 … = v K ≠ 0 w h e r e   v 1 , v 2 … , v K − 1 ∈ H ═ , Def . 11 C a s e   2 b : I f v 1 = v 2 … = v A , v A + 1 = v A + 2 … = v B … v K − K + 1 … = v K − 1 = v K w h e r e   v 1 , v 2 … , v K ∈ H ═ , Def .   12

We can immediately see from Case 2 that it is possible to have a finite number of subsets of non-trivial identical vectors exist in H ═ which is dependent on its cardinality (Kundu, 2023a). We will formally define the number of subsets that can be formed as τ (Def. (13)) (Kundu, 2023a), τ ≝ 1 + ∑ t = 2 t = # H ═ − 2 # H ═ t , 10 = 1   i f f   H ═ = 2,3 ≥ 2   i f f   H ═ ≥ 4 . 10.1

We define the exact number of vectors to be reassigned on account of their uniqueness as τ ═ , i.e., from H ═ to H (Def. (14)). We will now compute the number finite subsets for different cardinalities of H ═ ,

For τ = 1 , i f v 1 = v 2 … = v K | K = # H ═ = 2,3 , t h e n , v 1 ∨ v 2 … ∨ v K = ⋁ k = 1 k = K v k ∈ H a n d , v 1 , v 2 … , v K − 1 ∈ H ═ , Def . 15 C l e a r l y , τ ═ = 1 , (11) τ ⇒ # H = # H + τ , ═ (12) = # H + 1 , (12.1) a n d , # H ═ = A − 1 , (13) = # H ═ − 1 . (13.1)

For τ = 2 , the corresponding data is i f   v 1 = v 2 … = v A   a n d   v A + 1 = v A + 2 … = v K | K = # H ═ ≥ 4 , t h e n , v 1 ∨ v 2 … ∨ v A = ⋁ k = 1 k = A v k ∈ H a n d , v 1 , v 2 … , v A − 1 ∈ H ═   a n d v A + 1 ∨ v A + 2 … ∨ v K = ⋁ k = A + 1 k = K v k ∈ H a n d , v A + 1 ∨ v A + 2 … , v K − A − 1 ∈ H ═ . (Def. 16) H e r e , τ ═ A = τ ═ K = 1 , (14) ⇒ τ ═ ∼ τ A + τ K , (15) = 2 , (15.1) ⇒ # H = # H + τ , (16) = # H + 2 , (16.1) a n d   # H ═ = A − 1 + K − A − 1 , (17) = K − 2 , (17.1) = # H ═ − 2 . (17.2)

In general, for τ-subsets of identical vectors in H ═ the number of τ ═ -vectors that will be reassigned will be numerically identical ( τ ∼ τ ═ ), # H = # H + τ ═ , (18) # H ═ = # H ═ − τ ═ , (19)

“ReDirection” is freely available and can be updated or installed directly from the graphics user interface (GUI) (R-4.1. x) as “update.packages (‘ReDirection’)” and/or “install.packages (‘ReDirection’)” from any of the CRAN mirrors. “ReDirection” is built in RStudio (1.4.1717) and tested in R-4.1. x. “ReDirection” comprises three functions (calculate_reaction_vector, check_matrix, and reaction_vector). The dependencies for “ReDirection” are the packages “pracma,” “MASS,” “stats,” and the combinations function from the R-package (“gtools”). The downloaded package includes detailed documentation of all the functions, along with ready-to-use examples and tests of functionality. “ReDirection” utilizes these functions sequentially and processes the stoichiometry number matrix of the reactants/products and reactions of a biochemical network that is defined by the user (Figure 1). In addition to implementing “ReDirection” locally, several R-scripts are developed in house, and used to preformat (input and output) and analyze data. The algorithm followed by “ReDirection” can be divided into simpler steps. These include checking the user-defined stoichiometry matrix, searching for a suitable null space-generated subspace, screening and partitioning terms, and computing the probable dissociation constant (Figure 1).

It has already been proven that the algorithm deployed by “ReDirection” is likely to be NP-hard (Kundu, 2023a). This means that for a biochemical network whose output is determined by summing its constituent terms, there is a limit on the maximum number of reaction vectors that can be modeled by a user. Since “ReDirection” utilizes combinatorial summations to identify a suitable null space-generated subspace from where the probable reaction constants for a modeled biochemical network can be computed, the upper bound for the maximum number of reaction vectors is likely to be lower, i.e., there is a narrow permissible limit.

Since “ReDirection” needs to be user-friendly, an indicator of this must be available a priori. We utilize the time metric to ascertain this numerically. In other words, the time T   min that “ReDirection” takes to unambiguously annotate every reaction of a biochemical network is utilized to delineate an upper bound for the maximum number of reaction vectors that the user can incorporate for a biochemical network. We create several stoichiometry number matrices n = 50 in accordance with the previously established constraints and examine the run-time that “ReDirection” takes to compute the probable dissociation constants for the simulated yet plausible biochemical networks (Figure 2A; Supplementary Text S1) (Kundu, 2023a). The studies are carried out on a system with the following configuration: i5-10400F processor, clock speed 2.9 GHz, 64-bit, 16 GB RAM.

In order to assess these observations, we compute a truth table with the following assumptions and abbreviations (Defs 26–29): T P ∶ = T i m e   T − t o − u n a m b i g u o u s   a n n o t a t i o n   o f   r e a c t i o n   w h e r e   # V ∈ 2,4   a n d   T ∈ 0   min ⁡ , 20   min , F P ∶ = T i m e   T − t o − u n a m b i g u o u s   a n n o t a t i o n   o f   r e a c t i o n   w h e r e   # V > 4   a n d   T ∈ 0   min ⁡ , 20   min , F N ∶ = T i m e   T − t o − u n a m b i g u o u s   a n n o t a t i o n   o f   r e a c t i o n   w h e r e   # V ∈ 2,4   a n d   T > 20   min ⁡ , T N ∶ = T i m e   T − t o − u n a m b i g u o u s   a n n o t a t i o n   o f   r e a c t i o n   w h e r e   # V > 4   a n d   T > 20   min ⁡ .

This yields the following indices to assess our premise: P r e c i s i o n ≝ T P T P + F P , (49) R e c a l l   s e n s i t i v i t y ≝ T P T P + F N , (50) S p e c i f i c i t y ≝ 1 − F P F P + T N , (51) = T N T N + F N , (51.1) A c c u r a c y ≝ T P + T N T P + F P + F N + T N . (52)

We conclude this study by examining the relevance of the probable dissociation constants that are computed by “ReDirection” in physiologically relevant biochemical networks for galactose metabolism and heme and cholesterol biosynthesis. The stoichiometry number matrices for these networks are constructed in accordance with the numerical constraints discussed here and in previous work (Figure 3; Figure 4; Figure 5; Supplementary Texts S2–S4) (Kundu, 2023a; Kundu, 2023b).

Galactose–glucose interconversion is readily observed within the cell, catalyzed by the enzyme UDP-galactose 4-epimerase E C   5.1.3.2 , and suggests a biochemical network with several potentially bidirectional reactions (Conte et al., 2021; Nicoli et al., 2021). Here, “ReDirection” computes the probable dissociation constants for every reaction of a constrained biochemical network I ═ G a l = 13 , J G a l = 11 for human galactose metabolism (Eqs. (53, 54)) (Figure 3; Supplementary Text S2). The effect of perturbing r 1 − r 11 is investigated by introducing the atypical reactions r 12 (UDP-galactose → alpha-D-galactose 1-phosphate) and r 13 (UDP-galactose → alpha-D-galactose 1-phosphate) into the network (Figure 3). The enzymes (UTP-hexose 1-phosphate uridyltransferase, E C   2.7.7.10 ; UTP-monosaccharide-1-phosphate uridyltransferase, E C   2.7.7.64 ) that mediate the transformation of UDP-galactose to alpha-D-galactose 1-phosphate are not significant contributors to human galactose metabolism. This reaction is mediated by UDP-glucose-hexose-1-phosphate uridyltransferase ( E C   2.7.7.12 ) ( r 9 ) and is a major regulatory checkpoint for galactose–glucose interconversion (Figure 3). Cholesterol biosynthesis is the result of the mevalonate and non-mevalonate pathways, along with a well-characterized mitochondrial shunt pathway that may function to protect hydroxy-methyl-glutaryl (HMG) CoA reductase E C   1.1.1.34 from the deleterious effects of mevalonate (Edmond and Popjak, 1974; Nakanishi et al., 1988; Eisenreich et al., 2004; Buhaescu and Izzedine, 2007). Here, we present, analyze, and discuss a biochemical network I ═ C h o l e s = 18 , J C h o l e s = 15 for eukaryotic cholesterol synthesis by the mevalonate pathway, along with the shunt pathway (Eqs. (55, 56)) (Figure 4; Supplementary Text S3). Heme biosynthesis is central to the utilization of iron in the transport of oxygen and carbon dioxide via hemoglobin and other proteins, bilirubin-mediated conjugation and excretion of xenobiotics, and electron transfer in oxidative phosphorylation (Paoli et al., 2002; Thom et al., 2013; Poulos, 2014). We present, analyze, and discuss a biochemical network I ═ H e m e = 21 , J H e m e = 18 for heme biosynthesis and explore the effects of uroporphyrins (I) and (III) and coproporphyrins (I) and (III) on the immediate precursors uroporphyrinogens (I) and (III) or products coproporphyrinogens (I) and (III) (Eqs. (57, 58)) (Figure 5; Supplementary Text S4).

“ReDirection” utilizes the aforementioned functions sequentially and processes the stoichiometric number matrix for the biochemical network that is defined by the user and computes the probable dissociation constant for every reaction (Figure 1). This is conducted sequentially as follows:

Step 1

“ReDirection” checks whether the matrix of stoichiometry numbers that the user inputs is compliant with previously outlined criteria and does not have any linear dependent vectors. If found, “ReDirection” excludes them. The modified input matrix is rechecked.

Step 2

“ReDirection” then computes the null space of the checked/rechecked stoichiometric number matrix of the reactants/products and reactions of the user-defined biochemical network.

Step 3

“ReDirection” processes and screens this null space for redundant and/or trivial vectors and defines a subspace by excluding the same.

Step 4

“ReDirection” combinatorially sums the remaining vectors, i.e., non-redundant and non-trivial, and repeats step 3 for a finite number of u -iterations, where u = M ∈ N .

Step 5

For u = U > M iterations, “ReDirection” defines, populates, and computes several descriptors (sum, arithmetic mean, and standard deviation) for a reaction-specific sequence vector with terms that are drawn from each row of a null space-generated subspace.

Step 5a

“ReDirection” tests each term of an i t h -reaction-specific sequence vector for convergence.

Step 5b

If this term diverges and possesses a numerical value greater than 2 standard deviations from the mean, then this term is binned into the appropriate outcome-specific (forward/reverse/equivalent) subset.

Step 5c

The terms of each outcome-specific subset form a finite series whose sum is computed by “ReDirection.”

Step 5d

“ReDirection” then maps these sums to strictly positive real numbers which are then specific for each outcome-specific subset.

Step 5e

These outcome-specific numerical measures form the i t h -reaction-specific outcome vector.

Step 6

“ReDirection” computes the p1-norm of the reaction-specific outcome vector and annotates the reaction.

Step 7

“ReDirection” checks whether the annotations for all the other reactions of the user-defined biochemical network are unambiguous.

Step 7a

If there is no reaction that has been annotated ambiguously, then “ReDirection” outputs the predicted outcomes for every reaction of the user-defined biochemical network.

Step 7b

If there is a reaction that has been annotated ambiguously, then “ReDirection” continues the iterations.

Step 7c

“ReDirection” combinatorially sums all non-redundant and non-trivial null space-generated subspace vectors that remain, defines a new subspace, and repeats steps 5–7.

The data suggest that the cardinality of the null space-generated subspace that is chosen to compute the probable dissociation constant for a reaction determines not only the time taken to complete the computations but also whether this can be accomplished in real time (Figures 2B–D; Table 3). It was observed that this was achievable, i.e., T ∈ 0   min ⁡ , 20   min , regularly for null spaces with 2-, 3-, and 4-null space spanning vectors (Figures 2B–D; Table 3). However, when the nullity exceeded 4, the computations did not terminate even when the run time significantly exceeded 20 min T > 60   min (Table 2): T ∈ 0   min ⁡ , 20   min i f f   I ═ ″ R e D i r e c t i o n ″ ∈ J + 2 , J + 4 a n d   # V ∈ 2,4 , (59) T > 20   min i f   I ═ ″ R e D i r e c t i o n ″ > J + 4 a n d   # V > 4 . (60)

On the basis of the time taken by “ReDirection” to complete the annotations for each simulated biochemical network, we can categorize each outcome in terms of the categorical variables T P , F P , F N , T N (Table 3). The complete dataset is summarized as follows: T P = 26 , (61) F P = 0 , (62) F N = 11 , (63) T N = 13 . (64)

This yields the following indices to assess our premise: P r e c i s i o n = 100 % , (65) R e c a l l   s e n s i t i v i t y ≈ 70 % , (66) S p e c i f i c i t y = 100 % , (67) A c c u r a c y ≈ 78 % . (68)

Clearly, we can achieve significant proportioning of these data on the basis of our estimate of an upper bound for the reactions of these simulated biochemical networks (accuracy, precision, specificity, and recall). We suggest the following bounds for the number of reactions which a user may specify for a modeled biochemical network: I ═ ″ R e D i r e c t i o n ″ ∈ J + 2 , J + 4 . (69)

We now utilize “ReDirection” with these constraints to compute probable dissociation constants and, thence, investigate the biochemical networks of human galactose metabolism and cholesterol biosynthesis.

The presented biochemical network for galactose metabolism comprises a significantly larger fraction ≈ 63 % of equivalent reactions, as compared to the forward ≈ 27 % and reverse ≈ 10 % reactions (Figure 3). We also observe the directional preference of several reactions η 1 = η 2 ≈ 12.7 , η 3 ≈ 26 , η 10 ≈ 0.04 toward the synthesis of UDP-glucose (Figure 3). This, when coupled with the equivalent and sequential conversions to UDP-galactose, lactose, and galactose, η 4 − 6 ≈ 1.000 ensures that there is minimal change to the pool of galactose-containing complex carbohydrates and lipids (glycosphingolipids, gangliosides, cerebrosides, and mucopolysaccharides) (Thom et al., 2013; Poulos, 2014). Additionally, since the magnitude of the probable dissociation constant for η 3 is twice that of η 1 and η 2 η 3 η 1 = η 3 η 1 > 2.0 , the utilization of alpha-galactose 1-phosphate is faster than its synthesis. Here, in addition, by the law of mass action, there is a net flux of the network toward the biosynthesis of UDP-glucose (Figure 3). In this scenario, the atypical reactions r 12 , r 13 function to perturb galactose metabolism with flux toward the synthesis of galactose 1-phosphate η 12 ≈ 13.6 or galactose η 13 ≈ 12.7 from UDP-galactose and either complements or compensates, where applicable, reactions r 1 and r 2 (Figure 3). These studies suggest a predilection of the biochemical network toward synthesizing galactose, which, along with the activity of UDP-galactose 4-epimerase, constitute a plausible explanation for the milder clinical profile of the inborn errors of galactose metabolism (Raff et al., 1978; Jessen et al., 1985) (Figure 3).

The high number of equivalent reactions ≈ 44 % studied for cholesterol biosynthesis in the biochemical network suggests a regulatory role, which may account for the conservation of this pathway across taxa (eukaryotes, bacteria, and archaea) (Figure 4) (Edmond and Popjak, 1974; Nakanishi et al., 1988). Catabolism of the cyclopentanoperhydrophenanthrene (CPPP) ring of cholesterol is elaborate and occurs via the incorporation of a single molecule of oxygen by the heme- and iron-dependent cyclooxygenase P450 monooxygenase system of enzymes. The shunt pathway is a simple and yet an effective way of redirecting mevalonate prior to ring closure (Edmond and Popjak, 1974; Nakanishi et al., 1988; Pappu et al., 2002; Roullet et al., 2012). Smith–Lemli–Opitz syndrome is an inborn error of metabolism that arises due to mutations of the terminal enzyme in cholesterol biosynthesis (delta-7-reductase; E C   1.3.1.21 ) (Pappu et al., 2002; Roullet et al., 2012). This is postulated to result in an increased flux of mevalonate through the shunt pathway along with a concomitant increase in the excretion of urinary mevalonate (Edmond and Popjak, 1974; Nakanishi et al., 1988; Pappu et al., 2002; Roullet et al., 2012). This study supports this notion, at least in theory, with all the probable dissociation constants favoring a prominent role for the shunt pathway r 11 → r 8 → r 7 → r 17 η 11 ≈ 0.158 , η 8 ≈ 0.028 , η 7 ≈ 45 , η 17 ≈ 9.572 (Figure 4). The isomeric conversion of isopentenyl pyrophosphate to dimethylallyl pyrophosphate from mevalonate has the greatest numerical value of all the predicted probable dissociation constants η 7 ≈ 45 for the biochemical network. This, in addition, supports the rapid removal of mevalonate either by conversion (main, shunt) and/or excretion in urine (Edmond and Popjak, 1974; Nakanishi et al., 1988; Pappu et al., 2002; Roullet et al., 2012).

The distribution of equivalent ≈ 53 % ; I ═ H e m e = 11 , forward ≈ 37 % ; I ═ H e m e = 8 , and reverse ( ≈ 10 % ; I ═ H e m e = 2 ) reactions supports a similar inference for heme biosynthesis (Figure 5). The rate-limiting step for heme biosynthesis is the reaction catalyzed by ALAS1 ( δ -aminolevulinate synthetase I; E C   2.3.1.37 ). Our data suggest that the reactions from coproporphyrinogens (I) and (III) to uroporphyrinogens (I) and (III) η 7 ≈ η 10 ≈ 0.03 may also contribute significantly to this self-regulation. These reactions are catalyzed by uroporphyrinogen decarboxylase ( E C   4.1.1.37 ) and uroporphyrins (I) and (III), along with the coproporphyrins (I) and (III) that are subsequently generated by sunlight or the spontaneous removal of protons and can function as organic free radicals (Stein et al., 2017). Additionally, and in the absence of enzyme-catalyzed reactions, this catalytic sequestration of the substrates (uroporphyrinogens (I) and (III) and coproporphyrinogens (I) and (III)) ensures, by the law of mass action, that the flux is toward the biosynthesis of heme and its incorporation into heme proteins (Figure 5). Furthermore, since the free radical cycle involving these is self-propagating, the accumulated products, once generated, considerably damage the neighboring skin and other tissues. This reaction clinically partitions disorders of heme biosynthesis into those with photosensitivity and those with predominantly neuropsychiatric manifestations. Porphyria cutanea tarda (PCT) is an inborn error of metabolism due to a defect in the enzyme uroporphyrinogen decarboxylase and results in debilitating blisters on the skin due to exposure to sunlight (Stein et al., 2017). Our data η 7 ≈ η 10 ≈ 0.03 offer a plausible explanation for the genesis and pathophysiology of PCT (Figure 5).

The probable dissociation constants for a biochemical network provides the user with theoretically sound and biochemically relevant indices by which reactions of a biochemical network can be compared along with the corresponding change in the reactants/products (Reinker et al., 2006; Lecca et al., 2009; Haraldsdottir et al., 2012; Shindo et al., 2018; Wittenstein et al., 2022; Kundu, 2023a). A potentially novel application for these data is to incorporate these into simulation studies with the stochastic simulation algorithms (Gillespie, 2007; Kundu, 2016; Kundu, 2021). However, these studies mandate, by definition, the use of every possible reaction during a simulation run. This precludes the direct usage of data that are generated by “ReDirection” since only half the reactions are considered in computing the probable dissociation constants for the modeled biochemical network. The complete set of reactions for a user-defined biochemical network I ═ u s e r is simply I ═ u s e r = 2 . I ═ ″ R e D i r e c t i o n ″ . (70)

We annotate this set of additional half reactions in terms of the probable dissociation constant for the “ReDirection” annotated reaction as (Kundu, 2016; Kundu, 2021; Kundu, 2023a) i f   η i ∈ R ∩ 1 , ∞ R ∩ 0,1 R ∩ 1 , t h e n   η − i = 1.0 , (Def. 30) f o r   t h e   p a i r   o f   r e v e r s i b l e   r e a c t i o n s   r i , r − i .

This approach has yielded interesting insights into the export of high-affinity peptides to the plasma membrane by the major histocompatibility complex-I (MHC1) (Kundu, 2021). In that study, the authors examined a low-affinity peptide-driven biochemical network that could also be potentially regulatory and, therefore, important in priming circulating CD8⁺ T-cell lymphocytes into mounting a suitable immune response in the presence of acute and chronic insults (Kundu, 2021). Similarly, a role for reactive oxygen species in facilitating cellular proliferation and transmigration whilst precluding a cell to senescence and apoptosis concomitantly was addressed by creating a biochemical network for an advancing phagocyte toward a noxious stimulus (Kundu, 2016). The transduced signal was modeled to act through lipid raft-interacting actin fibers that could stabilize the actin cytoskeleton of the phagocyte and promote the development of a single dominant lamellipodium in the direction of the noxious stimulus (Kundu, 2016).